author | Scott Morrison <scott@tqft.net> |
Fri, 16 Jul 2010 13:23:07 -0600 | |
changeset 441 | c50ae482fe6a |
parent 438 | 0d62ea7c653d |
child 440 | 379e9a10c079 |
permissions | -rw-r--r-- |
94 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
3 |
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
|
199 | 4 |
\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
94 | 5 |
|
312 | 6 |
\section{$n$-categories and their modules} |
94 | 7 |
\label{sec:ncats} |
8 |
||
108 | 9 |
\subsection{Definition of $n$-categories} |
339
9698f584e732
starting to revise the ancient TQFTs-from-fields section; other minor stuff
Kevin Walker <kevin@canyon23.net>
parents:
336
diff
changeset
|
10 |
\label{ss:n-cat-def} |
108 | 11 |
|
94 | 12 |
Before proceeding, we need more appropriate definitions of $n$-categories, |
13 |
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
14 |
(As is the case throughout this paper, by ``$n$-category" we mean some notion of |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
15 |
a ``weak" $n$-category with ``strong duality".) |
94 | 16 |
|
141 | 17 |
The definitions presented below tie the categories more closely to the topology |
18 |
and avoid combinatorial questions about, for example, the minimal sufficient |
|
19 |
collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
20 |
For examples of topological origin |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
21 |
(e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
22 |
it is easy to show that they |
141 | 23 |
satisfy our axioms. |
24 |
For examples of a more purely algebraic origin, one would typically need the combinatorial |
|
25 |
results that we have avoided here. |
|
26 |
||
27 |
\medskip |
|
28 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
29 |
There are many existing definitions of $n$-categories, with various intended uses. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
30 |
In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
31 |
Generally, these sets are indexed by instances of a certain typical shape. |
347
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
32 |
Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on). |
94 | 33 |
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
34 |
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
|
35 |
and so on. |
|
36 |
(This allows for strict associativity.) |
|
263
fc3e10aa0d40
minor edits at the beginning of ncat
Scott Morrison <scott@tqft.net>
parents:
261
diff
changeset
|
37 |
Still other definitions (see, for example, \cite{MR2094071}) |
94 | 38 |
model the $k$-morphisms on more complicated combinatorial polyhedra. |
39 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
40 |
For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
41 |
Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
42 |
to the standard $k$-ball. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
43 |
By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
103 | 44 |
standard $k$-ball. |
45 |
We {\it do not} assume that it is equipped with a |
|
263
fc3e10aa0d40
minor edits at the beginning of ncat
Scott Morrison <scott@tqft.net>
parents:
261
diff
changeset
|
46 |
preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
103 | 47 |
|
109 | 48 |
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
49 |
the boundary), we want a corresponding |
|
94 | 50 |
bijection of sets $f:\cC(X)\to \cC(Y)$. |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
51 |
(This will imply ``strong duality", among other things.) Putting these together, we have |
94 | 52 |
|
187 | 53 |
\begin{axiom}[Morphisms] |
54 |
\label{axiom:morphisms} |
|
55 |
For each $0 \le k \le n$, we have a functor $\cC_k$ from |
|
103 | 56 |
the category of $k$-balls and |
187 | 57 |
homeomorphisms to the category of sets and bijections. |
58 |
\end{axiom} |
|
59 |
||
94 | 60 |
|
61 |
(Note: We usually omit the subscript $k$.) |
|
62 |
||
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
63 |
We are being deliberately vague about what flavor of $k$-balls |
195 | 64 |
we are considering. |
94 | 65 |
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
66 |
They could be topological or PL or smooth. |
|
195 | 67 |
%\nn{need to check whether this makes much difference} |
94 | 68 |
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
386 | 69 |
to be fussier about corners and boundaries.) |
94 | 70 |
For each flavor of manifold there is a corresponding flavor of $n$-category. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
71 |
For simplicity, we will concentrate on the case of PL unoriented manifolds. |
94 | 72 |
|
311
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
73 |
(The ambitious reader may want to keep in mind two other classes of balls. |
319
121c580d5ef7
editting all over the place
Scott Morrison <scott@tqft.net>
parents:
314
diff
changeset
|
74 |
The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
311
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
75 |
This will be used below to describe the blob complex of a fiber bundle with |
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
76 |
base space $Y$. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
77 |
The second is balls equipped with a section of the tangent bundle, or the frame |
311
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
78 |
bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
79 |
These can be used to define categories with less than the ``strong" duality we assume here, |
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
80 |
though we will not develop that idea fully in this paper.) |
62d112a2df12
mention some other flavors of balls
Kevin Walker <kevin@canyon23.net>
parents:
310
diff
changeset
|
81 |
|
94 | 82 |
Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
83 |
of morphisms). |
|
84 |
The 0-sphere is unusual among spheres in that it is disconnected. |
|
85 |
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
|
319
121c580d5ef7
editting all over the place
Scott Morrison <scott@tqft.net>
parents:
314
diff
changeset
|
86 |
(Actually, this is only true in the oriented case, with 1-morphisms parameterized |
359
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
356
diff
changeset
|
87 |
by {\it oriented} 1-balls.) |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
88 |
For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
89 |
For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
90 |
(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
91 |
We prefer to not make the distinction in the first place. |
263
fc3e10aa0d40
minor edits at the beginning of ncat
Scott Morrison <scott@tqft.net>
parents:
261
diff
changeset
|
92 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
93 |
Instead, we will combine the domain and range into a single entity which we call the |
94 | 94 |
boundary of a morphism. |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
95 |
Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
96 |
In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
97 |
$1\le k \le n$. |
313 | 98 |
At first it might seem that we need another axiom for this, but in fact once we have |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
99 |
all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
426
8aca80203f9d
search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/
Kevin Walker <kevin@canyon23.net>
parents:
425
diff
changeset
|
100 |
construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
101 |
to spheres (and any other manifolds): |
94 | 102 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
103 |
\begin{lem} |
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
104 |
\label{lem:spheres} |
333 | 105 |
For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
106 |
the category of $k{-}1$-spheres and |
187 | 107 |
homeomorphisms to the category of sets and bijections. |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
108 |
\end{lem} |
94 | 109 |
|
402 | 110 |
We postpone the proof of this result until after we've actually given all the axioms. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
111 |
Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
112 |
along with the data described in the other axioms at lower levels. |
263
fc3e10aa0d40
minor edits at the beginning of ncat
Scott Morrison <scott@tqft.net>
parents:
261
diff
changeset
|
113 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
114 |
%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
94 | 115 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
116 |
\begin{axiom}[Boundaries]\label{nca-boundary} |
333 | 117 |
For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
187 | 118 |
These maps, for various $X$, comprise a natural transformation of functors. |
119 |
\end{axiom} |
|
94 | 120 |
|
121 |
(Note that the first ``$\bd$" above is part of the data for the category, |
|
122 |
while the second is the ordinary boundary of manifolds.) |
|
123 |
||
333 | 124 |
Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
94 | 125 |
|
126 |
Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
|
103 | 127 |
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
333 | 128 |
all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
94 | 129 |
(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
130 |
\nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} |
94 | 131 |
and all the structure maps of the $n$-category should be compatible with the auxiliary |
132 |
category structure. |
|
133 |
Note that this auxiliary structure is only in dimension $n$; |
|
134 |
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
|
135 |
||
136 |
\medskip |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
137 |
|
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
138 |
(In order to simplify the exposition we have concentrated on the case of |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
139 |
unoriented PL manifolds and avoided the question of what exactly we mean by |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
140 |
the boundary a manifold with extra structure, such as an oriented manifold. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
141 |
In general, all manifolds of dimension less than $n$ should be equipped with the germ |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
142 |
of a thickening to dimension $n$, and this germ should carry whatever structure we have |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
143 |
on $n$-manifolds. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
144 |
In addition, lower dimensional manifolds should be equipped with a framing |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
145 |
of their normal bundle in the thickening; the framing keeps track of which |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
146 |
side (iterated) bounded manifolds lie on. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
147 |
For example, the boundary of an oriented $n$-ball |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
148 |
should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
149 |
bundle and a choice of direction in this bundle indicating |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
150 |
which side the $n$-ball lies on.) |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
151 |
|
94 | 152 |
\medskip |
153 |
||
154 |
We have just argued that the boundary of a morphism has no preferred splitting into |
|
155 |
domain and range, but the converse meets with our approval. |
|
156 |
That is, given compatible domain and range, we should be able to combine them into |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
157 |
the full boundary of a morphism. |
402 | 158 |
The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$ |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
159 |
on spheres. |
94 | 160 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
161 |
\begin{lem}[Boundary from domain and range] |
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
162 |
\label{lem:domain-and-range} |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
163 |
Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
164 |
$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
333 | 165 |
Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
166 |
two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
|
187 | 167 |
Then we have an injective map |
94 | 168 |
\[ |
402 | 169 |
\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
94 | 170 |
\] |
187 | 171 |
which is natural with respect to the actions of homeomorphisms. |
333 | 172 |
(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
173 |
becomes a normal product.) |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
174 |
\end{lem} |
94 | 175 |
|
179 | 176 |
\begin{figure}[!ht] |
186 | 177 |
$$ |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
178 |
\begin{tikzpicture}[%every label/.style={green} |
333 | 179 |
] |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
180 |
\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
181 |
\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
186 | 182 |
\draw (S) arc (-90:90:1); |
183 |
\draw (N) arc (90:270:1); |
|
184 |
\node[left] at (-1,1) {$B_1$}; |
|
185 |
\node[right] at (1,1) {$B_2$}; |
|
186 |
\end{tikzpicture} |
|
187 |
$$ |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
188 |
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
179 | 189 |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
190 |
Note that we insist on injectivity above. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
191 |
The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
109 | 192 |
|
333 | 193 |
Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
402 | 194 |
We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
109 | 195 |
|
195 | 196 |
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
333 | 197 |
as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
195 | 198 |
|
333 | 199 |
We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
110 | 200 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
333 | 201 |
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
195 | 202 |
More generally, we also include under the rubric ``restriction map" the |
203 |
the boundary maps of Axiom \ref{nca-boundary} above, |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
204 |
another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
205 |
of restriction maps. |
195 | 206 |
In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
207 |
($i = 1, 2$, notation from previous paragraph). |
|
208 |
These restriction maps can be thought of as |
|
209 |
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
|
94 | 210 |
|
211 |
||
212 |
Next we consider composition of morphisms. |
|
213 |
For $n$-categories which lack strong duality, one usually considers |
|
214 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
215 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
216 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
217 |
one general type of composition which can be in any ``direction". |
|
218 |
||
187 | 219 |
\begin{axiom}[Composition] |
220 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
|
179 | 221 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 222 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 223 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
224 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
225 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
226 |
We have a map |
94 | 227 |
\[ |
228 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
229 |
\] |
|
230 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
231 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
232 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
187 | 233 |
(For $k=n$, see below.) |
234 |
\end{axiom} |
|
94 | 235 |
|
179 | 236 |
\begin{figure}[!ht] |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
237 |
$$ |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
238 |
\begin{tikzpicture}[%every label/.style={green}, |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
239 |
x=1.5cm,y=1.5cm] |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
240 |
\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
241 |
\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
242 |
\draw (S) arc (-90:90:1); |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
243 |
\draw (N) arc (90:270:1); |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
244 |
\draw (N) -- (S); |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
245 |
\node[left] at (-1/4,1) {$B_1$}; |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
246 |
\node[right] at (1/4,1) {$B_2$}; |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
247 |
\node at (1/6,3/2) {$Y$}; |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
248 |
\end{tikzpicture} |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
249 |
$$ |
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
250 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
179 | 251 |
|
195 | 252 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 253 |
The composition (gluing) maps above are strictly associative. |
254 |
\end{axiom} |
|
102 | 255 |
|
423 | 256 |
\nn{should say this means $N$ at a time, not just 3 at a time} |
257 |
||
179 | 258 |
\begin{figure}[!ht] |
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
259 |
$$\mathfig{.65}{ncat/strict-associativity}$$ |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
260 |
\caption{An example of strict associativity.}\label{blah6}\end{figure} |
179 | 261 |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
262 |
We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. |
110 | 263 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
195 | 264 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
265 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
|
266 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
267 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 268 |
|
192 | 269 |
We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
270 |
We will call elements of $\cC(B)_Y$ morphisms which are |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
271 |
``splittable along $Y$'' or ``transverse to $Y$''. |
192 | 272 |
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
109 | 273 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
274 |
More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
193 | 275 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
276 |
the smaller balls to $X$. |
|
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
277 |
We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
193 | 278 |
In situations where the subdivision is notationally anonymous, we will write |
279 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
|
280 |
the unnamed subdivision. |
|
335
9bf409eb5040
mostly finished inserting \cl
Scott Morrison <scott@tqft.net>
parents:
334
diff
changeset
|
281 |
If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
193 | 282 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
283 |
subdivision of $\bd X$ and no competing subdivision of $X$. |
|
192 | 284 |
|
285 |
The above two composition axioms are equivalent to the following one, |
|
102 | 286 |
which we state in slightly vague form. |
287 |
||
288 |
\xxpar{Multi-composition:} |
|
289 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
290 |
into small $k$-balls, there is a |
|
291 |
map from an appropriate subset (like a fibered product) |
|
193 | 292 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 293 |
and these various $m$-fold composition maps satisfy an |
365
a93bb76a8525
moving an already prepared diagram out of tempkw
Scott Morrison <scott@tqft.net>
parents:
364
diff
changeset
|
294 |
operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
179 | 295 |
|
296 |
\begin{figure}[!ht] |
|
365
a93bb76a8525
moving an already prepared diagram out of tempkw
Scott Morrison <scott@tqft.net>
parents:
364
diff
changeset
|
297 |
$$\mathfig{.8}{ncat/operad-composition}$$ |
a93bb76a8525
moving an already prepared diagram out of tempkw
Scott Morrison <scott@tqft.net>
parents:
364
diff
changeset
|
298 |
\caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
95 | 299 |
|
300 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
301 |
||
343 | 302 |
\begin{axiom}[Product (identity) morphisms, preliminary version] |
303 |
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
|
304 |
usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
|
305 |
These maps must satisfy the following conditions. |
|
306 |
\begin{enumerate} |
|
307 |
\item |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
308 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram |
343 | 309 |
\[ \xymatrix{ |
310 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
|
311 |
X \ar[r]^{f} & X' |
|
312 |
} \] |
|
313 |
commutes, then we have |
|
314 |
\[ |
|
315 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
316 |
\] |
|
317 |
\item |
|
318 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
319 |
\[ |
|
320 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
321 |
\] |
|
322 |
and |
|
323 |
\[ |
|
324 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
325 |
\] |
|
326 |
\item |
|
327 |
Product morphisms are associative: |
|
328 |
\[ |
|
329 |
(a\times D)\times D' = a\times (D\times D') . |
|
330 |
\] |
|
331 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
332 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
333 |
\item |
|
334 |
Product morphisms are compatible with restriction: |
|
335 |
\[ |
|
336 |
\res_{X\times E}(a\times D) = a\times E |
|
337 |
\] |
|
338 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
339 |
\end{enumerate} |
|
340 |
\end{axiom} |
|
341 |
||
342 |
We will need to strengthen the above preliminary version of the axiom to allow |
|
343 |
for products which are ``pinched" in various ways along their boundary. |
|
352
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
344 |
(See Figure \ref{pinched_prods}.) |
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
345 |
\begin{figure}[t] |
364
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
346 |
$$ |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
347 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
348 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
349 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
350 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
351 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
352 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
353 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
354 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
355 |
\draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
356 |
\draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
357 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
358 |
\qquad \qquad |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
359 |
\begin{tikzpicture}[baseline=-0.15cm] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
360 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
361 |
\path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
362 |
\draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
363 |
\foreach \x in {-6, -5.5, ..., 0} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
364 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
365 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
366 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
367 |
\draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
368 |
\draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
369 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
370 |
$$ |
352
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
371 |
\caption{Examples of pinched products}\label{pinched_prods} |
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
372 |
\end{figure} |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
373 |
(The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
344 | 374 |
where we construct a traditional category from a topological category.) |
343 | 375 |
Define a {\it pinched product} to be a map |
376 |
\[ |
|
377 |
\pi: E\to X |
|
378 |
\] |
|
344 | 379 |
such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
343 | 380 |
on a standard iterated degeneracy map |
381 |
\[ |
|
344 | 382 |
d: \Delta^{k+m}\to\Delta^k . |
343 | 383 |
\] |
384 |
(We thank Kevin Costello for suggesting this approach.) |
|
385 |
||
344 | 386 |
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
343 | 387 |
and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
344 | 388 |
$l \le m$, with $l$ depending on $x$. |
343 | 389 |
|
390 |
It is easy to see that a composition of pinched products is again a pinched product. |
|
391 |
||
392 |
A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
|
393 |
$\pi:E'\to \pi(E')$ is again a pinched product. |
|
394 |
A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
|
395 |
such that each $E_i\sub E$ is a sub pinched product. |
|
352
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
396 |
(See Figure \ref{pinched_prod_unions}.) |
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
397 |
\begin{figure}[t] |
364
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
398 |
$$ |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
399 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
400 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
401 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
402 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
403 |
\draw[blue] (0,0) -- (5.66,0); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
404 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
405 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
406 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
407 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
408 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
409 |
\qquad |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
410 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
411 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
412 |
\path[clip] (0,-1) rectangle (4,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
413 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
414 |
\draw[blue] (0,0) -- (5,0); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
415 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
416 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
417 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
418 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
419 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
420 |
\qquad |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
421 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
422 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
423 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
424 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
425 |
\draw[blue] (2.83,3) circle (3); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
426 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
427 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
428 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
429 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
430 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
431 |
$$ |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
432 |
$$ |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
433 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
434 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
435 |
\path[clip] (0,-1) rectangle (4,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
436 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
437 |
\draw[blue] (0,-1) -- (4,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
438 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
439 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
440 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
441 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
442 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
443 |
\qquad |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
444 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
445 |
\begin{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
446 |
\path[clip] (0,-1) rectangle (5,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
447 |
\draw[blue,line width=2pt] (0,-1) rectangle (5,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
448 |
\draw[blue] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
449 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
450 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
451 |
} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
452 |
\end{scope} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
453 |
\end{tikzpicture} |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
454 |
$$ |
93d636f420c7
converting some hand drawn pictures to tikz
Scott Morrison <scott@tqft.net>
parents:
359
diff
changeset
|
455 |
\caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
352
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
348
diff
changeset
|
456 |
\end{figure} |
343 | 457 |
|
458 |
The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
|
459 |
$\pi:E\to X$. |
|
344 | 460 |
Morphisms in the image of $\pi^*$ will be called product morphisms. |
343 | 461 |
Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
462 |
In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
|
344 | 463 |
In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
464 |
define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
|
465 |
same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
|
343 | 466 |
|
467 |
||
468 |
\addtocounter{axiom}{-1} |
|
187 | 469 |
\begin{axiom}[Product (identity) morphisms] |
344 | 470 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
471 |
there is a map $\pi^*:\cC(X)\to \cC(E)$. |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
472 |
These maps must satisfy the following conditions. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
473 |
\begin{enumerate} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
474 |
\item |
344 | 475 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
476 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
95 | 477 |
\[ \xymatrix{ |
344 | 478 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
95 | 479 |
X \ar[r]^{f} & X' |
480 |
} \] |
|
109 | 481 |
commutes, then we have |
482 |
\[ |
|
344 | 483 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
109 | 484 |
\] |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
485 |
\item |
344 | 486 |
Product morphisms are compatible with gluing (composition). |
487 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
|
488 |
be pinched products with $E = E_1\cup E_2$. |
|
489 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
|
490 |
Then |
|
109 | 491 |
\[ |
344 | 492 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
109 | 493 |
\] |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
494 |
\item |
344 | 495 |
Product morphisms are associative. |
423 | 496 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
109 | 497 |
\[ |
344 | 498 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
109 | 499 |
\] |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
500 |
\item |
344 | 501 |
Product morphisms are compatible with restriction. |
502 |
If we have a commutative diagram |
|
503 |
\[ \xymatrix{ |
|
504 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
505 |
Y \ar@{^(->}[r] & X |
|
506 |
} \] |
|
507 |
such that $\rho$ and $\pi$ are pinched products, then |
|
110 | 508 |
\[ |
344 | 509 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
110 | 510 |
\] |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
511 |
\end{enumerate} |
187 | 512 |
\end{axiom} |
95 | 513 |
|
343 | 514 |
|
515 |
\medskip |
|
128 | 516 |
|
95 | 517 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
518 |
The last axiom (below), concerning actions of |
|
519 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
520 |
||
521 |
We start with the plain $n$-category case. |
|
522 |
||
420 | 523 |
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
187 | 524 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
95 | 525 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
187 | 526 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
267 | 527 |
\end{axiom} |
96 | 528 |
|
174 | 529 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 530 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 531 |
Let $J$ be a 1-ball (interval). |
532 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 533 |
(Here we use the ``pinched" version of $Y\times J$. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
534 |
\nn{do we need notation for this?}) |
96 | 535 |
We define a map |
536 |
\begin{eqnarray*} |
|
537 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
538 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
539 |
\end{eqnarray*} |
|
142 | 540 |
(See Figure \ref{glue-collar}.) |
189 | 541 |
\begin{figure}[!ht] |
542 |
\begin{equation*} |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
543 |
\begin{tikzpicture} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
544 |
\def\rad{1} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
545 |
\def\srad{0.75} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
546 |
\def\gap{4.5} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
547 |
\foreach \i in {0, 1, 2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
548 |
\node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
549 |
\node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
550 |
\foreach \n in {1,2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
551 |
\fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
552 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
553 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
554 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
555 |
\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
556 |
\draw[decorate] (0-intersection-1.east) -- (0-intersection-2.east); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
557 |
\end{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
558 |
\node[right=1mm] at (0.east) {$a$}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
559 |
\draw[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
560 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
561 |
\draw (1-small) circle (\srad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
562 |
\foreach \theta in {90, 72, ..., -90} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
563 |
\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
564 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
565 |
\filldraw[fill=white] (1) circle (\rad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
566 |
\foreach \n in {1,2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
567 |
\fill (intersection \n of 1-small and 1) circle (2pt); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
568 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
569 |
\node[below] at (1-small.south) {$a \times J$}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
570 |
\draw[->] ($(1.east)+(1,0)$) -- ($(2.west)+(-0.2,0)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
571 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
572 |
\begin{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
573 |
\path[clip] (2) circle (\rad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
574 |
\draw[clip] (2.east) circle (\srad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
575 |
\foreach \y in {1, 0.86, ..., -1} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
576 |
\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
577 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
578 |
\end{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
579 |
\end{tikzpicture} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
580 |
\end{equation*} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
581 |
\begin{equation*} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
582 |
\xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} |
189 | 583 |
\end{equation*} |
584 |
||
585 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
586 |
We call a map of this form a {\it collar map}. |
96 | 587 |
It can be thought of as the action of the inverse of |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
588 |
a map which projects a collar neighborhood of $Y$ onto $Y$, |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
589 |
or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$ |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
590 |
to a larger collar. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
591 |
We call the equivalence relation generated by collar maps and homeomorphisms |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
592 |
isotopic (rel boundary) to the identity {\it extended isotopy}. |
96 | 593 |
|
594 |
The revised axiom is |
|
595 |
||
267 | 596 |
\addtocounter{axiom}{-1} |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
597 |
\begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
187 | 598 |
\label{axiom:extended-isotopies} |
599 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
600 |
to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
187 | 601 |
Then $f$ acts trivially on $\cC(X)$. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
602 |
In addition, collar maps act trivially on $\cC(X)$. |
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
603 |
\end{axiom} |
96 | 604 |
|
97 | 605 |
\smallskip |
606 |
||
607 |
For $A_\infty$ $n$-categories, we replace |
|
608 |
isotopy invariance with the requirement that families of homeomorphisms act. |
|
609 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
|
416 | 610 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
611 |
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
|
612 |
||
97 | 613 |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
614 |
\addtocounter{axiom}{-1} |
420 | 615 |
\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
335
9bf409eb5040
mostly finished inserting \cl
Scott Morrison <scott@tqft.net>
parents:
334
diff
changeset
|
616 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
97 | 617 |
\[ |
618 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
619 |
\] |
|
620 |
These action maps are required to be associative up to homotopy |
|
621 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
437 | 622 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
97 | 623 |
\nn{repeat diagram here?} |
187 | 624 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
625 |
\end{axiom} |
97 | 626 |
|
416 | 627 |
We should strengthen the above axiom to apply to families of collar maps. |
628 |
To do this we need to explain how collar maps form a topological space. |
|
629 |
Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
|
97 | 630 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
416 | 631 |
Having chains on the space of collar maps act gives rise to coherence maps involving |
632 |
weak identities. |
|
420 | 633 |
We will not pursue this in detail here. |
97 | 634 |
|
103 | 635 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
636 |
into a plain $n$-category (enriched over graded groups). |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
637 |
In a different direction, if we enrich over topological spaces instead of chain complexes, |
97 | 638 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
639 |
instead of $C_*(\Homeo_\bd(X))$. |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
640 |
Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
97 | 641 |
type $A_\infty$ $n$-category. |
642 |
||
99 | 643 |
\medskip |
97 | 644 |
|
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
645 |
The alert reader will have already noticed that our definition of a (plain) $n$-category |
416 | 646 |
is extremely similar to our definition of a system of fields. |
647 |
There are two differences. |
|
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
648 |
First, for the $n$-category definition we restrict our attention to balls |
99 | 649 |
(and their boundaries), while for fields we consider all manifolds. |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
650 |
Second, in category definition we directly impose isotopy |
416 | 651 |
invariance in dimension $n$, while in the fields definition we |
652 |
instead remember a subspace of local relations which contain differences of isotopic fields. |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
653 |
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
654 |
Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
655 |
balls and, at level $n$, quotienting out by the local relations: |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
656 |
\begin{align*} |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
657 |
\cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
658 |
\end{align*} |
142 | 659 |
This $n$-category can be thought of as the local part of the fields. |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
660 |
Conversely, given a topological $n$-category we can construct a system of fields via |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
661 |
a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 662 |
|
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
663 |
\subsection{Examples of $n$-categories} |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
664 |
\label{ss:ncat-examples} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
665 |
|
101 | 666 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
667 |
We now describe several classes of examples of $n$-categories satisfying our axioms. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
668 |
We typically specify only the morphisms; the rest of the data for the category |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
669 |
(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
101 | 670 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
671 |
\begin{example}[Maps to a space] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
672 |
\rm |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
673 |
\label{ex:maps-to-a-space}% |
425 | 674 |
Let $T$ be a topological space. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
675 |
We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
676 |
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
677 |
all continuous maps from $X$ to $T$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
678 |
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 679 |
homotopies fixed on $\bd X$. |
101 | 680 |
(Note that homotopy invariance implies isotopy invariance.) |
681 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
682 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
683 |
\end{example} |
313 | 684 |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
685 |
\noop{ |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
686 |
Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
687 |
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
688 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
689 |
an n-cat} |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
690 |
} |
101 | 691 |
|
423 | 692 |
\begin{example}[Maps to a space, with a fiber] \label{ex:maps-with-fiber} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
693 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
694 |
\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 695 |
We can modify the example above, by fixing a |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
696 |
closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
697 |
otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
698 |
Taking $F$ to be a point recovers the previous case. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
699 |
\end{example} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
700 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
701 |
\begin{example}[Linearized, twisted, maps to a space] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
702 |
\rm |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
703 |
\label{ex:linearized-maps-to-a-space}% |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
704 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 705 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
706 |
(have in mind the trivial cocycle). |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
707 |
For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
708 |
For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
709 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 710 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
711 |
$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
712 |
(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
713 |
Alternatively, we could equip the balls with fundamental classes.) |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
714 |
\end{example} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
715 |
|
425 | 716 |
\begin{example}[$n$-categories from TQFTs] |
717 |
\rm |
|
718 |
\label{ex:ncats-from-tqfts}% |
|
719 |
Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
|
720 |
system of fields (also denoted $\cF$) and local relations. |
|
721 |
Let $W$ be an $n{-}j$-manifold. |
|
722 |
Define the $j$-category $\cF(W)$ as follows. |
|
723 |
If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
|
724 |
If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
|
725 |
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
|
726 |
\end{example} |
|
727 |
||
728 |
The next example is only intended to be illustrative, as we don't specify |
|
729 |
which definition of a ``traditional $n$-category" we intend. |
|
730 |
Further, most of these definitions don't even have an agreed-upon notion of |
|
731 |
``strong duality", which we assume here. |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
732 |
\begin{example}[Traditional $n$-categories] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
733 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
734 |
\label{ex:traditional-n-categories} |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
735 |
Given a ``traditional $n$-category with strong duality" $C$ |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
736 |
define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
737 |
to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
339
9698f584e732
starting to revise the ancient TQFTs-from-fields section; other minor stuff
Kevin Walker <kevin@canyon23.net>
parents:
336
diff
changeset
|
738 |
For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
739 |
combinations of $C$-labeled embedded cell complexes of $X$ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
740 |
modulo the kernel of the evaluation map. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
741 |
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
742 |
with each cell labelled according to the corresponding cell for $a$. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
743 |
(These two cells have the same codimension.) |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
744 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
745 |
Define $\cC(X)$, for $\dim(X) < n$, |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
746 |
to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
747 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
748 |
to be the dual Hilbert space $A(X\times F; c)$. |
426
8aca80203f9d
search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/
Kevin Walker <kevin@canyon23.net>
parents:
425
diff
changeset
|
749 |
(See \S\ref{sec:constructing-a-tqft}.) |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
750 |
\end{example} |
313 | 751 |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
752 |
\noop{ |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
753 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
754 |
an n-cat} |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
755 |
Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
756 |
$C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
757 |
\nn{KW: We already refer to \S \ref{sec:fields} above} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
758 |
Constructing a system of fields from $\cC$ recovers that example. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
759 |
\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
760 |
\nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
761 |
where the quotient is built in. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
762 |
but (string diagrams)/(relations) is isomorphic to |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
763 |
(pasting diagrams composed of smaller string diagrams)/(relations)} |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
764 |
} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
765 |
|
204 | 766 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
767 |
\newcommand{\Bord}{\operatorname{Bord}} |
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
768 |
\begin{example}[The bordism $n$-category, plain version] |
348
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
769 |
\label{ex:bord-cat} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
770 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
771 |
\label{ex:bordism-category} |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
772 |
For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
773 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
196 | 774 |
to $\bd X$. |
225
32a76e8886d1
minor tweaks on small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
224
diff
changeset
|
775 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
776 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 777 |
$W \to W'$ which restricts to the identity on the boundary. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
778 |
\end{example} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
779 |
|
196 | 780 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 781 |
|
196 | 782 |
%\begin{example}[Variation on the above examples] |
783 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
784 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
785 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
786 |
%%to think of these guys as affording a representation |
|
787 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
788 |
%\end{example} |
|
101 | 789 |
|
790 |
||
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
791 |
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 792 |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
793 |
\begin{example}[Chains (or space) of maps to a space] |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
794 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
795 |
\label{ex:chains-of-maps-to-a-space} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
796 |
We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
797 |
For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
798 |
Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
799 |
\[ |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
800 |
C_*(\Maps_c(X\times F \to T)), |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
801 |
\] |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
802 |
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 803 |
and $C_*$ denotes singular chains. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
804 |
Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
805 |
we get an $A_\infty$ $n$-category enriched over spaces. |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
806 |
\end{example} |
101 | 807 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
808 |
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
809 |
homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
810 |
|
279 | 811 |
\begin{example}[Blob complexes of balls (with a fiber)] |
812 |
\rm |
|
813 |
\label{ex:blob-complexes-of-balls} |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
814 |
Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
291 | 815 |
We will define an $A_\infty$ $k$-category $\cC$. |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
816 |
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
291 | 817 |
When $X$ is an $k$-ball, |
279 | 818 |
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
819 |
where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
|
820 |
\end{example} |
|
101 | 821 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
399
diff
changeset
|
822 |
This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
823 |
Notice that with $F$ a point, the above example is a construction turning a topological |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
824 |
$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
825 |
We think of this as providing a ``free resolution" |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
826 |
of the topological $n$-category. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
827 |
\nn{say something about cofibrant replacements?} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
828 |
In fact, there is also a trivial, but mostly uninteresting, way to do this: |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
829 |
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
830 |
and take $\CD{B}$ to act trivially. |
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
831 |
|
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
832 |
Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
833 |
It's easy to see that with $n=0$, the corresponding system of fields is just |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
834 |
linear combinations of connected components of $T$, and the local relations are trivial. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
835 |
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
836 |
|
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
837 |
\begin{example}[The bordism $n$-category, $A_\infty$ version] |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
838 |
\rm |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
839 |
\label{ex:bordism-category-ainf} |
348
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
840 |
As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
841 |
to be the set of all $k$-dimensional |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
842 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
843 |
to $\bd X$. |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
844 |
For an $n$-ball $X$ with boundary condition $c$ |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
845 |
define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
846 |
submanifolds $W$ of $X\times \Real^\infty$ such that |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
847 |
$W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
848 |
(The topology on this space is induced by ambient isotopy rel boundary. |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
849 |
This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
850 |
$W'$ runs though representatives of homeomorphism types of such manifolds.) |
b2fab3bf491b
A-inf bordism cat example
Kevin Walker <kevin@canyon23.net>
parents:
347
diff
changeset
|
851 |
\nn{check this} |
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
852 |
\end{example} |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
853 |
|
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
854 |
|
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
855 |
|
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
856 |
Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
857 |
copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
858 |
(We require that the interiors of the little balls be disjoint, but their |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
859 |
boundaries are allowed to meet. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
860 |
Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
861 |
the embeddings of a ``little" ball with image all of the big ball $B^n$. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
862 |
\nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
863 |
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
864 |
by shrinking the little balls (precomposing them with dilations), |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
865 |
we see that both operads are homotopic to the space of $k$ framed points |
401
a8b8ebcf07ac
Making notation in the product theorem more consistent.
Scott Morrison <scott@tqft.net>
parents:
400
diff
changeset
|
866 |
in $B^n$. |
a8b8ebcf07ac
Making notation in the product theorem more consistent.
Scott Morrison <scott@tqft.net>
parents:
400
diff
changeset
|
867 |
It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
868 |
an action of $\cE\cB_n$. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
869 |
\nn{add citation for this operad if we can find one} |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
870 |
|
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
871 |
\begin{example}[$E_n$ algebras] |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
872 |
\rm |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
873 |
\label{ex:e-n-alg} |
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
874 |
|
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
875 |
Let $A$ be an $\cE\cB_n$-algebra. |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
876 |
Note that this implies a $\Diff(B^n)$ action on $A$, |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
877 |
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
878 |
We will define an $A_\infty$ $n$-category $\cC^A$. |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
879 |
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
880 |
In other words, the $k$-morphisms are trivial for $k<n$. |
347
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
881 |
If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
882 |
(Plain colimit, not homotopy colimit.) |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
883 |
Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
884 |
the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
885 |
embedded balls into a single larger embedded ball. |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
886 |
To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
887 |
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
888 |
Alternatively and more simply, we could define $\cC^A(X)$ to be |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
889 |
$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
890 |
The remaining data for the $A_\infty$ $n$-category |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
891 |
--- composition and $\Diff(X\to X')$ action --- |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
892 |
also comes from the $\cE\cB_n$ action on $A$. |
14643c4931bc
finished E_n example (at SFO)
Kevin Walker <kevin@canyon23.net>
parents:
346
diff
changeset
|
893 |
\nn{should we spell this out?} |
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
894 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
895 |
\nn{Should remark that the associated hocolim for manifolds |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
896 |
is agrees with Lurie's topological chiral homology construction; maybe wait |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
897 |
until next subsection to say that?} |
356
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
898 |
|
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
899 |
Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
900 |
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
901 |
an $\cE\cB_n$-algebra. |
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
902 |
\nn{The paper is already long; is it worth giving details here?} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
903 |
\end{example} |
95 | 904 |
|
108 | 905 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
906 |
\subsection{From balls to manifolds} |
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
907 |
\label{ss:ncat_fields} \label{ss:ncat-coend} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
908 |
In this section we describe how to extend an $n$-category $\cC$ as described above |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
909 |
(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
910 |
This extension is a certain colimit, and we've chosen the notation to remind you of this. |
402 | 911 |
Thus we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
912 |
from $k$-balls to arbitrary $k$-manifolds. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
913 |
Recall that we've already anticipated this construction in the previous section, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
914 |
inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
915 |
so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
916 |
In the case of plain $n$-categories, this construction factors into a construction of a |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
917 |
system of fields and local relations, followed by the usual TQFT definition of a |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
918 |
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
919 |
For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
920 |
Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
921 |
an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
922 |
We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
923 |
for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
108 | 924 |
|
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
925 |
We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
926 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
927 |
and we will define $\cl{\cC}(W)$ as a suitable colimit |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
928 |
(or homotopy colimit in the $A_\infty$ case) of this functor. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
929 |
We'll later give a more explicit description of this colimit. |
420 | 930 |
In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
931 |
then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
108 | 932 |
|
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
933 |
Define a {\it permissible decomposition} of $W$ to be a cell decomposition |
108 | 934 |
\[ |
935 |
W = \bigcup_a X_a , |
|
936 |
\] |
|
142 | 937 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
938 |
\nn{need to define this more carefully} |
108 | 939 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
940 |
of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
941 |
|
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
942 |
\begin{defn} |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
943 |
The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
944 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
945 |
See Figure \ref{partofJfig} for an example. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
946 |
\end{defn} |
119 | 947 |
|
948 |
\begin{figure}[!ht] |
|
949 |
\begin{equation*} |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
950 |
\mathfig{.63}{ncat/zz2} |
119 | 951 |
\end{equation*} |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
952 |
\caption{A small part of $\cell(W)$} |
119 | 953 |
\label{partofJfig} |
954 |
\end{figure} |
|
955 |
||
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
956 |
An $n$-category $\cC$ determines |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
957 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
108 | 958 |
(possibly with additional structure if $k=n$). |
197 | 959 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
960 |
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
|
961 |
are splittable along this decomposition. |
|
108 | 962 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
963 |
\begin{defn} |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
964 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
965 |
For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
966 |
\begin{equation} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
967 |
\label{eq:psi-C} |
197 | 968 |
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
969 |
\end{equation} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
970 |
where the restrictions to the various pieces of shared boundaries amongst the cells |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
971 |
$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
972 |
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
973 |
\end{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
974 |
|
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
975 |
If $k=n$ in the above definition and we are enriching in some auxiliary category, |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
976 |
we need to say a bit more. |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
977 |
We can rewrite Equation \ref{eq:psi-C} as |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
978 |
\begin{equation} \label{eq:psi-CC} |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
979 |
\psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
980 |
\end{equation} |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
981 |
where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
982 |
(which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
983 |
means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
984 |
If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
985 |
$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
986 |
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
987 |
|
420 | 988 |
Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
989 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
990 |
\begin{defn}[System of fields functor] |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
991 |
\label{def:colim-fields} |
402 | 992 |
If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
993 |
That is, for each decomposition $x$ there is a map |
402 | 994 |
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
995 |
above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
996 |
\end{defn} |
112 | 997 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
998 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
402 | 999 |
When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1000 |
is defined as above, as the colimit of $\psi_{\cC;W}$. |
402 | 1001 |
When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1002 |
\end{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1003 |
|
402 | 1004 |
We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1005 |
with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
111 | 1006 |
|
422 | 1007 |
We now give more concrete descriptions of the above colimits. |
1008 |
||
1009 |
In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
|
1010 |
the colimit is |
|
1011 |
\[ |
|
1012 |
\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim , |
|
1013 |
\] |
|
1014 |
where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
|
1015 |
induced by refinement and gluing. |
|
1016 |
If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
|
1017 |
we can take |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1018 |
\begin{equation*} |
422 | 1019 |
\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1020 |
\end{equation*} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1021 |
where $K$ is the vector space spanned by elements $a - g(a)$, with |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1022 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1023 |
\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1024 |
|
225
32a76e8886d1
minor tweaks on small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
224
diff
changeset
|
1025 |
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 1026 |
is more involved. |
142 | 1027 |
%\nn{should probably rewrite this to be compatible with some standard reference} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1028 |
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1029 |
Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
402 | 1030 |
Define $\cl{\cC}(W)$ as a vector space via |
112 | 1031 |
\[ |
402 | 1032 |
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 1033 |
\] |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1034 |
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1035 |
(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1036 |
the complex $U[m]$ is concentrated in degree $m$.) |
422 | 1037 |
\nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?} |
402 | 1038 |
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 1039 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
1040 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
402 | 1041 |
summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
112 | 1042 |
\[ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1043 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 1044 |
\] |
1045 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
|
198 | 1046 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
422 | 1047 |
%\nn{need to say this better} |
1048 |
%\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
1049 |
%combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
1050 |
%of $A_\infty$ category} |
|
108 | 1051 |
|
113 | 1052 |
We will call $m$ the filtration degree of the complex. |
422 | 1053 |
\nn{is there a more standard term for this?} |
113 | 1054 |
We can think of this construction as starting with a disjoint copy of a complex for each |
1055 |
permissible decomposition (filtration degree 0). |
|
1056 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
1057 |
(filtration degree 1). |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1058 |
Then we kill the extra homology we just introduced with mapping |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1059 |
cylinders between the mapping cylinders (filtration degree 2), and so on. |
113 | 1060 |
|
422 | 1061 |
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
108 | 1062 |
|
420 | 1063 |
It is easy to see that |
422 | 1064 |
there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
108 | 1065 |
comprise a natural transformation of functors. |
1066 |
||
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1067 |
\begin{lem} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1068 |
\label{lem:colim-injective} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1069 |
Let $W$ be a manifold of dimension less than $n$. Then for each |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1070 |
decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1071 |
\end{lem} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1072 |
\begin{proof} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1073 |
\nn{...} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1074 |
\end{proof} |
402 | 1075 |
|
108 | 1076 |
\nn{need to finish explaining why we have a system of fields; |
1077 |
define $k$-cat $\cC(\cdot\times W)$} |
|
1078 |
||
1079 |
\subsection{Modules} |
|
95 | 1080 |
|
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1081 |
Next we define plain and $A_\infty$ $n$-category modules. |
199 | 1082 |
The definition will be very similar to that of $n$-categories, |
1083 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
|
198 | 1084 |
|
104 | 1085 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 1086 |
in the context of an $m{+}1$-dimensional TQFT. |
1087 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
1088 |
This will be explained in more detail as we present the axioms. |
|
1089 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1090 |
Throughout, we fix an $n$-category $\cC$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1091 |
For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1092 |
We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
102 | 1093 |
|
1094 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1095 |
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
102 | 1096 |
We call $B$ the ball and $N$ the marking. |
1097 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
1098 |
restricts to a homeomorphism of markings. |
|
1099 |
||
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1100 |
\begin{module-axiom}[Module morphisms] |
102 | 1101 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
1102 |
the category of marked $k$-balls and |
|
1103 |
homeomorphisms to the category of sets and bijections.} |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1104 |
\end{module-axiom} |
102 | 1105 |
|
1106 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
1107 |
||
423 | 1108 |
For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1109 |
of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$. |
|
104 | 1110 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1111 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
423 | 1112 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$ |
1113 |
(see Example \ref{ex:maps-with-fiber}). |
|
104 | 1114 |
(The union is along $N\times \bd W$.) |
423 | 1115 |
%(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1116 |
%the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 1117 |
|
182 | 1118 |
\begin{figure}[!ht] |
224
9faf1f7fad3e
fixing signs in small blobs lemma
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
222
diff
changeset
|
1119 |
$$\mathfig{.8}{ncat/boundary-collar}$$ |
182 | 1120 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1121 |
||
103 | 1122 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1123 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 1124 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1125 |
\begin{lem} |
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1126 |
\label{lem:hemispheres} |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1127 |
{For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
104 | 1128 |
the category of marked $k$-hemispheres and |
102 | 1129 |
homeomorphisms to the category of sets and bijections.} |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1130 |
\end{lem} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1131 |
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1132 |
We use the same type of colimit construction. |
102 | 1133 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1134 |
In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
104 | 1135 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1136 |
\begin{module-axiom}[Module boundaries (maps)] |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1137 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
102 | 1138 |
These maps, for various $M$, comprise a natural transformation of functors.} |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1139 |
\end{module-axiom} |
102 | 1140 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1141 |
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 1142 |
|
1143 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
1144 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
1145 |
and $c\in \cC(\bd M)$. |
|
1146 |
||
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1147 |
\begin{lem}[Boundary from domain and range] |
423 | 1148 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1149 |
$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
|
104 | 1150 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1151 |
two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
423 | 1152 |
Then we have an injective map |
102 | 1153 |
\[ |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1154 |
\gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
102 | 1155 |
\] |
1156 |
which is natural with respect to the actions of homeomorphisms.} |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1157 |
\end{lem} |
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1158 |
Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
102 | 1159 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1160 |
Let $\cl\cM(H)_E$ denote the image of $\gl_E$. |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1161 |
We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
110 | 1162 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1163 |
\begin{lem}[Module to category restrictions] |
103 | 1164 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1165 |
$\cl\cM(H)\to \cC(H)$. |
103 | 1166 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1167 |
These maps comprise a natural transformation of functors.} |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1168 |
\end{lem} |
102 | 1169 |
|
103 | 1170 |
Note that combining the various boundary and restriction maps above |
110 | 1171 |
(for both modules and $n$-categories) |
103 | 1172 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1173 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 1174 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
1175 |
cutting submanifolds). |
|
103 | 1176 |
This fact will be used below. |
102 | 1177 |
|
104 | 1178 |
In our example, the various restriction and gluing maps above come from |
1179 |
restricting and gluing maps into $T$. |
|
1180 |
||
1181 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 1182 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 1183 |
(See Figure \ref{zzz3}.) |
103 | 1184 |
|
119 | 1185 |
\begin{figure}[!ht] |
1186 |
\begin{equation*} |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1187 |
\mathfig{.4}{ncat/zz3} |
119 | 1188 |
\end{equation*} |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1189 |
\caption{Module composition (top); $n$-category action (bottom).} |
119 | 1190 |
\label{zzz3} |
1191 |
\end{figure} |
|
1192 |
||
1193 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 1194 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1195 |
\begin{module-axiom}[Module composition] |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1196 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
103 | 1197 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1198 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
1199 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
1200 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
1201 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
1202 |
Then (axiom) we have a map |
|
1203 |
\[ |
|
1204 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
1205 |
\] |
|
1206 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1207 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
1208 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
1209 |
(For $k=n$, see below.)} |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1210 |
\end{module-axiom} |
119 | 1211 |
|
1212 |
||
103 | 1213 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
1214 |
module morphism. |
|
1215 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
1216 |
||
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1217 |
\begin{module-axiom}[$n$-category action] |
103 | 1218 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1219 |
$X$ is a plain $k$-ball, |
|
1220 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
1221 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
1222 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
1223 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
1224 |
Then (axiom) we have a map |
|
1225 |
\[ |
|
1226 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
1227 |
\] |
|
1228 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1229 |
to the intersection of the boundaries of $X$ and $M'$. |
|
1230 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
1231 |
(For $k=n$, see below.)} |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1232 |
\end{module-axiom} |
103 | 1233 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1234 |
\begin{module-axiom}[Strict associativity] |
423 | 1235 |
The composition and action maps above are strictly associative. |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1236 |
\end{module-axiom} |
103 | 1237 |
|
423 | 1238 |
\nn{should say that this is multifold, not just 3-fold} |
1239 |
||
110 | 1240 |
Note that the above associativity axiom applies to mixtures of module composition, |
1241 |
action maps and $n$-category composition. |
|
119 | 1242 |
See Figure \ref{zzz1b}. |
1243 |
||
1244 |
\begin{figure}[!ht] |
|
1245 |
\begin{equation*} |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1246 |
\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
119 | 1247 |
\end{equation*} |
1248 |
\caption{Two examples of mixed associativity} |
|
1249 |
\label{zzz1b} |
|
1250 |
\end{figure} |
|
1251 |
||
110 | 1252 |
|
1253 |
The above three axioms are equivalent to the following axiom, |
|
103 | 1254 |
which we state in slightly vague form. |
1255 |
\nn{need figure for this} |
|
1256 |
||
1257 |
\xxpar{Module multi-composition:} |
|
1258 |
{Given any decomposition |
|
1259 |
\[ |
|
1260 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
1261 |
\] |
|
1262 |
of a marked $k$-ball $M$ |
|
1263 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
1264 |
map from an appropriate subset (like a fibered product) |
|
1265 |
of |
|
1266 |
\[ |
|
1267 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
1268 |
\] |
|
1269 |
to $\cM(M)$, |
|
1270 |
and these various multifold composition maps satisfy an |
|
1271 |
operad-type strict associativity condition.} |
|
1272 |
||
423 | 1273 |
The above operad-like structure is analogous to the swiss cheese operad |
1274 |
\cite{MR1718089}. |
|
1275 |
||
1276 |
\medskip |
|
1277 |
||
1278 |
We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the |
|
1279 |
plain ball case. |
|
1280 |
Note that a marked pinched product can be decomposed into either |
|
1281 |
two marked pinched products or a plain pinched product and a marked pinched product. |
|
1282 |
\nn{should give figure} |
|
103 | 1283 |
|
423 | 1284 |
\begin{module-axiom}[Product (identity) morphisms] |
1285 |
For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
|
1286 |
$k{+}m$-ball ($m\ge 1$), |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1287 |
there is a map $\pi^*:\cM(M)\to \cM(E)$. |
423 | 1288 |
These maps must satisfy the following conditions. |
1289 |
\begin{enumerate} |
|
1290 |
\item |
|
1291 |
If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and |
|
1292 |
if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
103 | 1293 |
\[ \xymatrix{ |
423 | 1294 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
103 | 1295 |
M \ar[r]^{f} & M' |
1296 |
} \] |
|
423 | 1297 |
commutes, then we have |
1298 |
\[ |
|
1299 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
|
1300 |
\] |
|
1301 |
\item |
|
1302 |
Product morphisms are compatible with module composition and module action. |
|
1303 |
Let $\pi:E\to M$, $\pi_1:E_1\to M_1$, and $\pi_2:E_2\to M_2$ |
|
1304 |
be pinched products with $E = E_1\cup E_2$. |
|
1305 |
Let $a\in \cM(M)$, and let $a_i$ denote the restriction of $a$ to $M_i\sub M$. |
|
1306 |
Then |
|
1307 |
\[ |
|
1308 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
|
1309 |
\] |
|
1310 |
Similarly, if $\rho:D\to X$ is a pinched product of plain balls and |
|
1311 |
$E = D\cup E_1$, then |
|
1312 |
\[ |
|
1313 |
\pi^*(a) = \rho^*(a')\bullet \pi_1^*(a_1), |
|
1314 |
\] |
|
1315 |
where $a'$ is the restriction of $a$ to $D$. |
|
1316 |
\item |
|
1317 |
Product morphisms are associative. |
|
1318 |
If $\pi:E\to M$ and $\rho:D\to E$ are marked pinched products then |
|
1319 |
\[ |
|
1320 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
|
1321 |
\] |
|
1322 |
\item |
|
1323 |
Product morphisms are compatible with restriction. |
|
1324 |
If we have a commutative diagram |
|
1325 |
\[ \xymatrix{ |
|
1326 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
1327 |
Y \ar@{^(->}[r] & M |
|
1328 |
} \] |
|
1329 |
such that $\rho$ and $\pi$ are pinched products, then |
|
1330 |
\[ |
|
1331 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
|
1332 |
\] |
|
1333 |
($Y$ could be either a marked or plain ball.) |
|
1334 |
\end{enumerate} |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1335 |
\end{module-axiom} |
103 | 1336 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1337 |
As in the $n$-category definition, once we have product morphisms we can define |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1338 |
collar maps $\cM(M)\to \cM(M)$. |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1339 |
Note that there are two cases: |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1340 |
the collar could intersect the marking of the marked ball $M$, in which case |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1341 |
we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1342 |
in which case we use a product on a morphism of $\cC$. |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1343 |
|
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1344 |
In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1345 |
$a$ along a map associated to $\pi$. |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1346 |
|
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1347 |
\medskip |
110 | 1348 |
|
103 | 1349 |
There are two alternatives for the next axiom, according whether we are defining |
1350 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
1351 |
In the plain case we require |
|
1352 |
||
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1353 |
\begin{module-axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$] |
103 | 1354 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1355 |
to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
103 | 1356 |
Then $f$ acts trivially on $\cM(M)$.} |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1357 |
In addition, collar maps act trivially on $\cM(M)$. |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1358 |
\end{module-axiom} |
103 | 1359 |
|
1360 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
1361 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
1362 |
on $\bd B \setmin N$. |
|
1363 |
||
1364 |
For $A_\infty$ modules we require |
|
1365 |
||
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1366 |
\addtocounter{module-axiom}{-1} |
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1367 |
\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1368 |
For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
103 | 1369 |
\[ |
1370 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
1371 |
\] |
|
1372 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
1373 |
which fix $\bd M$. |
|
437 | 1374 |
These action maps are required to be associative up to homotopy, as in Theorem \ref{thm:CH-associativity}, |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1375 |
and also compatible with composition (gluing) in the sense that |
437 | 1376 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1377 |
\end{module-axiom} |
103 | 1378 |
|
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1379 |
As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1380 |
|
103 | 1381 |
\medskip |
102 | 1382 |
|
104 | 1383 |
Note that the above axioms imply that an $n$-category module has the structure |
1384 |
of an $n{-}1$-category. |
|
1385 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
346
90e0c5e7ae07
EB_n operad example; other misc stuff
Kevin Walker <kevin@canyon23.net>
parents:
344
diff
changeset
|
1386 |
where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
104 | 1387 |
above the non-marked boundary component of $J$. |
200 | 1388 |
(More specifically, we collapse $X\times P$ to a single point, where |
1389 |
$P$ is the non-marked boundary component of $J$.) |
|
104 | 1390 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 1391 |
|
105 | 1392 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1393 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
1394 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
1395 |
of marked 1-balls, call them left-marked and right-marked, |
|
1396 |
and hence there are two types of modules, call them right modules and left modules. |
|
1397 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
1398 |
there is no left/right module distinction. |
|
1399 |
||
130 | 1400 |
\medskip |
1401 |
||
224
9faf1f7fad3e
fixing signs in small blobs lemma
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
222
diff
changeset
|
1402 |
We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
9faf1f7fad3e
fixing signs in small blobs lemma
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
222
diff
changeset
|
1403 |
|
225
32a76e8886d1
minor tweaks on small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
224
diff
changeset
|
1404 |
\begin{example}[Examples from TQFTs] |
425 | 1405 |
\rm |
1406 |
Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
|
1407 |
and $\cF(W)$ the $j$-category associated to $W$. |
|
1408 |
Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
|
1409 |
Define a $\cF(W)$ module $\cF(Y)$ as follows. |
|
1410 |
If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
|
1411 |
$\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
|
1412 |
If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
|
1413 |
$\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
|
225
32a76e8886d1
minor tweaks on small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
224
diff
changeset
|
1414 |
\end{example} |
108 | 1415 |
|
224
9faf1f7fad3e
fixing signs in small blobs lemma
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
222
diff
changeset
|
1416 |
\begin{example} |
425 | 1417 |
\rm |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1418 |
Suppose $S$ is a topological space, with a subspace $T$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1419 |
We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1420 |
for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1421 |
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1422 |
such maps modulo homotopies fixed on $\bdy B \setminus N$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1423 |
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
420 | 1424 |
\end{example} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1425 |
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1426 |
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1427 |
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
224
9faf1f7fad3e
fixing signs in small blobs lemma
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
222
diff
changeset
|
1428 |
|
324
a20e2318cbb0
rewrite proof from gluing thm
Kevin Walker <kevin@canyon23.net>
parents:
319
diff
changeset
|
1429 |
\subsection{Modules as boundary labels (colimits for decorated manifolds)} |
112 | 1430 |
\label{moddecss} |
108 | 1431 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1432 |
Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1433 |
Let $W$ be a $k$-manifold ($k\le n$), |
143 | 1434 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1435 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
1436 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1437 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1438 |
the one appearing in \S \ref{ss:ncat_fields} above. |
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1439 |
(If $k = n$ and our $n$-categories are enriched, then |
108 | 1440 |
$\cC(W, \cN)$ will have additional structure; see below.) |
1441 |
||
1442 |
Define a permissible decomposition of $W$ to be a decomposition |
|
1443 |
\[ |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1444 |
W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
108 | 1445 |
\] |
435 | 1446 |
where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and |
1447 |
each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$, |
|
143 | 1448 |
with $M_{ib}\cap Y_i$ being the marking. |
1449 |
(See Figure \ref{mblabel}.) |
|
435 | 1450 |
\begin{figure}[t] |
1451 |
\begin{equation*} |
|
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1452 |
\mathfig{.4}{ncat/mblabel} |
435 | 1453 |
\end{equation*} |
1454 |
\caption{A permissible decomposition of a manifold |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1455 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
435 | 1456 |
Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel} |
1457 |
\end{figure} |
|
108 | 1458 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1459 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1460 |
This defines a partial ordering $\cell(W)$, which we will think of as a category. |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1461 |
(The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
108 | 1462 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1463 |
||
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1464 |
The collection of modules $\cN$ determines |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1465 |
a functor $\psi_\cN$ from $\cell(W)$ to the category of sets |
108 | 1466 |
(possibly with additional structure if $k=n$). |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1467 |
For a decomposition $x = (X_a, M_{ib})$ in $\cell(W)$, define $\psi_\cN(x)$ to be the subset |
108 | 1468 |
\[ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1469 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 1470 |
\] |
1471 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
1472 |
$X_a$ and $M_{ib}$ all agree. |
|
435 | 1473 |
(That is, the fibered product over the boundary restriction maps.) |
108 | 1474 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1475 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
1476 |
||
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1477 |
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
435 | 1478 |
(As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever |
1479 |
category we are enriching over, and if additionally we are in the $A_\infty$ case, |
|
1480 |
then we use a homotopy colimit.) |
|
1481 |
||
1482 |
\medskip |
|
108 | 1483 |
|
143 | 1484 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1485 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1486 |
$D\times Y_i \sub \bd(D\times W)$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1487 |
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
435 | 1488 |
has the structure of an $n{-}k$-category. |
144 | 1489 |
|
1490 |
\medskip |
|
1491 |
||
1492 |
We will use a simple special case of the above |
|
1493 |
construction to define tensor products |
|
1494 |
of modules. |
|
1495 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1496 |
(If $k=1$ and our manifolds are oriented, then one should be |
144 | 1497 |
a left module and the other a right module.) |
1498 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1499 |
Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
435 | 1500 |
$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1501 |
This of course depends (functorially) |
144 | 1502 |
on the choice of 1-ball $J$. |
105 | 1503 |
|
144 | 1504 |
We will define a more general self tensor product (categorified coend) below. |
1505 |
||
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1506 |
|
291 | 1507 |
\subsection{Morphisms of $A_\infty$ $1$-category modules} |
288 | 1508 |
\label{ss:module-morphisms} |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1509 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1510 |
In order to state and prove our version of the higher dimensional Deligne conjecture |
426
8aca80203f9d
search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/
Kevin Walker <kevin@canyon23.net>
parents:
425
diff
changeset
|
1511 |
(\S\ref{sec:deligne}), |
291 | 1512 |
we need to define morphisms of $A_\infty$ $1$-category modules and establish |
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1513 |
some of their elementary properties. |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1514 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1515 |
To motivate the definitions which follow, consider algebras $A$ and $B$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1516 |
right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1517 |
\begin{eqnarray*} |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1518 |
\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1519 |
f &\mapsto& [x \mapsto f(x\ot -)] \\ |
279 | 1520 |
{}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1521 |
\end{eqnarray*} |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1522 |
If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1523 |
\[ |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1524 |
(X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1525 |
\] |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1526 |
We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1527 |
and modules $\cM_\cC$ and $_\cC\cN$, |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1528 |
\[ |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1529 |
(\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1530 |
\] |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1531 |
|
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1532 |
In the next few paragraphs we define the objects appearing in the above equation: |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1533 |
$\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1534 |
$\hom_\cC$. |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1535 |
|
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1536 |
|
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1537 |
\def\olD{{\overline D}} |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1538 |
\def\cbar{{\bar c}} |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1539 |
In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1540 |
for general $n$. |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1541 |
For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1542 |
and their gluings (antirefinements). |
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1543 |
(This tensor product depends functorially on the choice of $J$.) |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1544 |
To a subdivision $D$ |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1545 |
\[ |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1546 |
J = I_1\cup \cdots\cup I_p |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1547 |
\] |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1548 |
we associate the chain complex |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1549 |
\[ |
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1550 |
\psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1551 |
\] |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1552 |
To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1553 |
module actions of $\cC$ on $\cM$ and $\cN$. |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1554 |
The underlying graded vector space of the homotopy colimit is |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1555 |
\[ |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1556 |
\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1557 |
\] |
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1558 |
where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1559 |
runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1560 |
We will denote an element of the summand indexed by $\olD$ by |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1561 |
$\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1562 |
The boundary map is given by |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1563 |
\begin{align*} |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1564 |
\bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
291 | 1565 |
& \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ |
1566 |
& \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n |
|
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1567 |
\end{align*} |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1568 |
where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1569 |
boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1570 |
and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1571 |
|
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1572 |
$(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1573 |
\[ |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1574 |
\prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1575 |
\] |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1576 |
where $(\psi(D_0)[l])^*$ denotes the linear dual. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1577 |
The boundary is given by |
291 | 1578 |
\begin{align} |
1579 |
\label{eq:tensor-product-boundary} |
|
1580 |
(-1)^{\deg f +1} (\bd f)(\olD\ot m\ot\cbar\ot n) & = f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) + f((\bd_+ \olD)\ot m\ot\cbar\ot n) + \\ |
|
1581 |
& \qquad + (-1)^{l} f(\olD\ot\bd m\ot\cbar \ot n) + (-1)^{l + \deg m} f(\olD\ot m\ot\bd \cbar \ot n) + \notag \\ |
|
1582 |
& \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
|
1583 |
\end{align} |
|
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1584 |
|
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1585 |
Next we define the dual module $(_\cC\cN)^*$. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1586 |
This will depend on a choice of interval $J$, just as the tensor product did. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1587 |
Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1588 |
to chain complexes. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1589 |
Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1590 |
\[ |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1591 |
(_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1592 |
\] |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1593 |
where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1594 |
to the right-marked interval $J\setmin K$. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1595 |
This extends to a functor from all left-marked intervals (not just those contained in $J$). |
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1596 |
\nn{need to say more here; not obvious how homeomorphisms act} |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1597 |
It's easy to verify the remaining module axioms. |
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1598 |
|
260 | 1599 |
Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1600 |
as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
260 | 1601 |
Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1602 |
Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
291 | 1603 |
Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1604 |
Then for each such $\olD$ we have a degree $l$ map |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1605 |
\begin{eqnarray*} |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1606 |
\cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1607 |
m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1608 |
\end{eqnarray*} |
260 | 1609 |
|
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1610 |
We are almost ready to give the definition of morphisms between arbitrary modules |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1611 |
$\cX_\cC$ and $\cY_\cC$. |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1612 |
Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
286
ff867bfc8e9c
mostly minor changes, reading modules section, stopping for dinner\!
Scott Morrison <scott@tqft.net>
parents:
279
diff
changeset
|
1613 |
To fix this, we define subdivisions as antirefinements of left-marked intervals. |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1614 |
Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1615 |
the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1616 |
omitted. |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1617 |
More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1618 |
gluing subintervals together and/or omitting some of the rightmost subintervals. |
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1619 |
(See Figure \ref{fig:lmar}.) |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1620 |
\begin{figure}[t]$$ |
381
84bcc5fdf8c2
experiment with tikz colors
Kevin Walker <kevin@canyon23.net>
parents:
367
diff
changeset
|
1621 |
\definecolor{arcolor}{rgb}{.75,.4,.1} |
386 | 1622 |
\begin{tikzpicture}[line width=1pt] |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1623 |
\fill (0,0) circle (.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1624 |
\draw (0,0) -- (2,0); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1625 |
\draw (1,0.1) -- (1,-0.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1626 |
|
381
84bcc5fdf8c2
experiment with tikz colors
Kevin Walker <kevin@canyon23.net>
parents:
367
diff
changeset
|
1627 |
\draw [->, arcolor] (1,0.25) -- (1,0.75); |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1628 |
|
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1629 |
\fill (0,1) circle (.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1630 |
\draw (0,1) -- (2,1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1631 |
\end{tikzpicture} |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1632 |
\qquad |
386 | 1633 |
\begin{tikzpicture}[line width=1pt] |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1634 |
\fill (0,0) circle (.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1635 |
\draw (0,0) -- (2,0); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1636 |
\draw (1,0.1) -- (1,-0.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1637 |
|
381
84bcc5fdf8c2
experiment with tikz colors
Kevin Walker <kevin@canyon23.net>
parents:
367
diff
changeset
|
1638 |
\draw [->, arcolor] (1,0.25) -- (1,0.75); |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1639 |
|
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1640 |
\fill (0,1) circle (.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1641 |
\draw (0,1) -- (1,1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1642 |
\end{tikzpicture} |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1643 |
\qquad |
386 | 1644 |
\begin{tikzpicture}[line width=1pt] |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1645 |
\fill (0,0) circle (.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1646 |
\draw (0,0) -- (3,0); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1647 |
\foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1648 |
\draw (\x,0.1) -- (\x,-0.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1649 |
} |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1650 |
|
381
84bcc5fdf8c2
experiment with tikz colors
Kevin Walker <kevin@canyon23.net>
parents:
367
diff
changeset
|
1651 |
\draw [->, arcolor] (1,0.25) -- (1,0.75); |
366
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1652 |
|
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1653 |
\fill (0,1) circle (.1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1654 |
\draw (0,1) -- (2,1); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1655 |
\foreach \x in {1.0, 1.5} { |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1656 |
\draw (\x,1.1) -- (\x,0.9); |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1657 |
} |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1658 |
|
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1659 |
\end{tikzpicture} |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1660 |
$$ |
b69b09d24049
tikzing left-marked-antirefinements
Scott Morrison <scott@tqft.net>
parents:
365
diff
changeset
|
1661 |
\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1662 |
|
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1663 |
Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1664 |
The underlying vector space is |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1665 |
\[ |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1666 |
\prod_l \prod_{\olD} \hom[l]\left( |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1667 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1668 |
\cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1669 |
\] |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1670 |
where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1671 |
(but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1672 |
$\hom[l](- \to -)$ means graded linear maps of degree $l$. |
260 | 1673 |
|
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1674 |
\nn{small issue (pun intended): |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1675 |
the above is a vector space only if the class of subdivisions is a set, e.g. only if |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1676 |
all of our left-marked intervals are contained in some universal interval (like $J$ above). |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1677 |
perhaps we should give another version of the definition in terms of natural transformations of functors.} |
260 | 1678 |
|
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1679 |
Abusing notation slightly, we will denote elements of the above space by $g$, with |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1680 |
\[ |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1681 |
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1682 |
\] |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1683 |
For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1684 |
where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1685 |
$\cbar''$ corresponds to the subintervals |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1686 |
which are dropped off the right side. |
386 | 1687 |
(If no such subintervals are dropped, then $\cbar''$ is empty.) |
291 | 1688 |
Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1689 |
we have |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1690 |
\begin{eqnarray*} |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1691 |
(\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
330 | 1692 |
& & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1693 |
\end{eqnarray*} |
291 | 1694 |
\nn{put in signs, rearrange terms to match order in previous formulas} |
330 | 1695 |
Here $\gl''$ denotes the module action in $\cY_\cC$ |
1696 |
and $\gl'$ denotes the module action in $\cX_\cC$. |
|
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1697 |
This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
260 | 1698 |
|
261
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1699 |
Note that if $\bd g = 0$, then each |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1700 |
\[ |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1701 |
g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1702 |
\] |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1703 |
constitutes a null homotopy of |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1704 |
$g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1705 |
should be interpreted as above). |
1c408505c9f5
finished def of module morphisms; still need to define (yet another) 'evaluation' map
Kevin Walker <kevin@canyon23.net>
parents:
260
diff
changeset
|
1706 |
|
410 | 1707 |
Define a {\it strong morphism} |
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1708 |
of modules to be a collection of {\it chain} maps |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1709 |
\[ |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1710 |
h_K : \cX(K)\to \cY(K) |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1711 |
\] |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1712 |
for each left-marked interval $K$. |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1713 |
These are required to commute with gluing; |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1714 |
for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1715 |
\[ \xymatrix{ |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1716 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1717 |
\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1718 |
\ar[d]^{\gl} \\ |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1719 |
\cX(K) \ar[r]^{h_{K}} & \cY(K) |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1720 |
} \] |
410 | 1721 |
Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. |
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1722 |
Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1723 |
If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1724 |
\[ |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1725 |
g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1726 |
\] |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1727 |
Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1728 |
If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1729 |
If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1730 |
that each $h_K$ is a chain map. |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1731 |
|
330 | 1732 |
We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ |
1733 |
as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. |
|
1734 |
\nn{ideally should give explicit examples of this in low degrees, |
|
1735 |
but skip that for now.} |
|
1736 |
\nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe |
|
1737 |
should make some arbitrary choice} |
|
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1738 |
\medskip |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1739 |
|
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1740 |
Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1741 |
\[ |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1742 |
g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1743 |
\] |
386 | 1744 |
\nn{...} |
1745 |
More generally, we have a chain map |
|
1746 |
\[ |
|
1747 |
\hom_\cC(\cX_\cC \to \cY_\cC) \ot \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
|
1748 |
\] |
|
330 | 1749 |
|
1750 |
\nn{not sure whether to do low degree examples or try to state the general case; ideally both, |
|
1751 |
but maybe just low degrees for now.} |
|
1752 |
||
1753 |
||
1754 |
\nn{...} |
|
262
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1755 |
|
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1756 |
|
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1757 |
|
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1758 |
|
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1759 |
\medskip |
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1760 |
|
3278eafef668
done for the moment with module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
261
diff
changeset
|
1761 |
|
330 | 1762 |
\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
1763 |
of the $\cC$ functors which commute with gluing only up to higher morphisms? |
|
1764 |
perhaps worth having both definitions available. |
|
1765 |
certainly the simple kind (strictly commute with gluing) arise in nature.} |
|
258
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1766 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1767 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1768 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1769 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1770 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1771 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1772 |
|
fd5d1647f4f3
starting write up module morphism def
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
1773 |
|
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1774 |
\subsection{The $n{+}1$-category of sphere modules} |
218 | 1775 |
\label{ssec:spherecat} |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1776 |
|
205 | 1777 |
In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
327 | 1778 |
whose objects are $n$-categories. |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1779 |
When $n=2$ |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1780 |
this is closely related to the familiar $2$-category of algebras, bimodules and intertwiners. |
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1781 |
While it is appropriate to call an $S^0$ module a bimodule, |
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1782 |
this is much less true for higher dimensional spheres, |
327 | 1783 |
so we prefer the term ``sphere module" for the general case. |
144 | 1784 |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
1785 |
The results of this subsection are not needed for the rest of the paper, |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1786 |
so we will skimp on details in a couple of places. We have included this mostly for the sake of comparing our notion of a topological $n$-category to other definitions. |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
1787 |
|
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
1788 |
For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
1789 |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
1790 |
The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
205 | 1791 |
these first. |
259
db18f7c32abe
more module morphism stuff
Kevin Walker <kevin@canyon23.net>
parents:
258
diff
changeset
|
1792 |
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1793 |
of $1$-category modules associated to decorated $n$-balls. |
205 | 1794 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1795 |
the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
205 | 1796 |
that our $n$-categories and modules have non-degenerate inner products. |
1797 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
1798 |
||
1799 |
\medskip |
|
1800 |
||
1801 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
|
1802 |
These will be defined in terms of certain classes of marked balls, very similarly |
|
1803 |
to the definition of $n$-category modules above. |
|
1804 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
1805 |
Because of this similarity, we only sketch the definitions below. |
|
1806 |
||
327 | 1807 |
We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
205 | 1808 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1809 |
Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
327 | 1810 |
$(B^k, B^{k-1})$. |
209 | 1811 |
See Figure \ref{feb21a}. |
205 | 1812 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1813 |
||
209 | 1814 |
\begin{figure}[!ht] |
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
1815 |
$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
209 | 1816 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
1817 |
\label{feb21a} |
|
1818 |
\end{figure} |
|
1819 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1820 |
The $0$-marked balls can be cut into smaller balls in various ways. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1821 |
We only consider those decompositions in which the smaller balls are either |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1822 |
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1823 |
or plain (don't intersect the $0$-marking of the large ball). |
327 | 1824 |
We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
205 | 1825 |
|
1826 |
Fix $n$-categories $\cA$ and $\cB$. |
|
327 | 1827 |
These will label the two halves of a $0$-marked $k$-ball. |
205 | 1828 |
|
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1829 |
An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is a collection of functors $\cM_k$ from the category |
327 | 1830 |
of $0$-marked $k$-balls, $1\le k \le n$, |
205 | 1831 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1832 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
327 | 1833 |
Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
205 | 1834 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1835 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
1836 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
416
diff
changeset
|
1837 |
Corresponding to this decomposition we have a composition (or ``gluing") map |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1838 |
from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
205 | 1839 |
|
1840 |
\medskip |
|
107 | 1841 |
|
327 | 1842 |
Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
206 | 1843 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1844 |
of $\cA$ and $\cB$. |
|
1845 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
1846 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
1847 |
\[ |
|
1848 |
\cD(X) \deq \cM(X\times J) . |
|
1849 |
\] |
|
1850 |
The product is pinched over the boundary of $J$. |
|
327 | 1851 |
The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
209 | 1852 |
(see Figure \ref{feb21b}). |
206 | 1853 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 1854 |
|
209 | 1855 |
\begin{figure}[!ht] |
367 | 1856 |
$$ |
1857 |
\begin{tikzpicture}[blue,line width=2pt] |
|
1858 |
\draw (0,1) -- (0,-1) node[below] {$X$}; |
|
1859 |
||
1860 |
\draw (2,0) -- (4,0) node[below] {$J$}; |
|
1861 |
\fill[red] (3,0) circle (0.1); |
|
1862 |
||
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
1863 |
\draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); |
367 | 1864 |
\draw[red] (top.center) -- (bottom.center); |
1865 |
\fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
|
1866 |
\fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
|
1867 |
||
1868 |
\path (bottom) node[below]{$X \times J$}; |
|
1869 |
||
1870 |
\end{tikzpicture} |
|
1871 |
$$ |
|
209 | 1872 |
\caption{The pinched product $X\times J$} |
1873 |
\label{feb21b} |
|
1874 |
\end{figure} |
|
1875 |
||
206 | 1876 |
More generally, consider an interval with interior marked points, and with the complements |
1877 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
1878 |
by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
|
209 | 1879 |
(See Figure \ref{feb21c}.) |
426
8aca80203f9d
search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/
Kevin Walker <kevin@canyon23.net>
parents:
425
diff
changeset
|
1880 |
To this data we can apply the coend construction as in \S\ref{moddecss} above |
327 | 1881 |
to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1882 |
This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
|
205 | 1883 |
|
209 | 1884 |
\begin{figure}[!ht] |
367 | 1885 |
$$ |
1886 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
1887 |
\draw[blue] (0,0) -- (6,0); |
|
1888 |
\foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
|
1889 |
\path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
|
1890 |
} |
|
1891 |
\foreach \x/\n in {1/0,2/1,4/2,5/3} { |
|
1892 |
\fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
|
1893 |
} |
|
1894 |
\end{tikzpicture} |
|
1895 |
\qquad |
|
1896 |
\qquad |
|
1897 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
1898 |
\draw[blue] (0,0) circle (2); |
|
1899 |
\foreach \q/\n in {-45/0,90/1,180/2} { |
|
1900 |
\path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
|
1901 |
} |
|
1902 |
\foreach \q/\n in {60/0,120/1,-120/2} { |
|
1903 |
\fill[red] (\q:2) circle (0.1); |
|
1904 |
\path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
|
1905 |
} |
|
1906 |
\end{tikzpicture} |
|
1907 |
$$ |
|
209 | 1908 |
\caption{Marked and labeled 1-manifolds} |
1909 |
\label{feb21c} |
|
1910 |
\end{figure} |
|
1911 |
||
206 | 1912 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1913 |
associated to the marked and labeled circle. |
|
209 | 1914 |
(See Figure \ref{feb21c}.) |
206 | 1915 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
327 | 1916 |
as the 2-sided tensor product of the two bimodules associated to the two intervals. |
206 | 1917 |
|
1918 |
\medskip |
|
1919 |
||
1920 |
Next we define $n$-category 1-sphere modules. |
|
1921 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
1922 |
circles (1-spheres) which we just introduced. |
|
1923 |
||
1924 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
1925 |
Fix a marked (and labeled) circle $S$. |
|
209 | 1926 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
207 | 1927 |
\nn{I need to make up my mind whether marked things are always labeled too. |
1928 |
For the time being, let's say they are.} |
|
1929 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
|
1930 |
where $B^j$ is the standard $j$-ball. |
|
399 | 1931 |
A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1932 |
(a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
|
1933 |
(See Figure xxxx.) |
|
207 | 1934 |
We now proceed as in the above module definitions. |
1935 |
||
209 | 1936 |
\begin{figure}[!ht] |
367 | 1937 |
$$ |
1938 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
1939 |
\draw[blue][fill=blue!15!white] (0,0) circle (2); |
367 | 1940 |
\fill[red] (0,0) circle (0.1); |
1941 |
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
1942 |
\draw[red] (0,0) -- (\qm:2); |
|
1943 |
\path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
1944 |
\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
1945 |
\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
1946 |
} |
|
1947 |
\end{tikzpicture} |
|
1948 |
$$ |
|
209 | 1949 |
\caption{Cone on a marked circle} |
1950 |
\label{feb21d} |
|
1951 |
\end{figure} |
|
1952 |
||
207 | 1953 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1954 |
\[ |
|
1955 |
\cD(X) \deq \cM(X\times C(S)) . |
|
1956 |
\] |
|
1957 |
The product is pinched over the boundary of $C(S)$. |
|
1958 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
1959 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
1960 |
||
1961 |
More generally, consider a 2-manifold $Y$ |
|
1962 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
1963 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
1964 |
the edges of $K$ are labeled by 0-sphere modules, |
|
1965 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
1966 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1967 |
If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-category |
207 | 1968 |
associated to the (marked, labeled) boundary of $Y$. |
1969 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
1970 |
||
1971 |
\medskip |
|
1972 |
||
1973 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
1974 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 1975 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 1976 |
|
1977 |
\medskip |
|
1978 |
||
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
1979 |
We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1980 |
Choose some collection of $n$-categories, then choose some collections of bimodules between |
207 | 1981 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
1982 |
possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
|
1983 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
|
1984 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
1985 |
There is a wide range of possibilities. |
|
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1986 |
The set $L_0$ could contain infinitely many $n$-categories or just one. |
207 | 1987 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1988 |
it could contain several. |
|
208 | 1989 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1990 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
1991 |
||
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
1992 |
We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
208 | 1993 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
1994 |
by elements of $L_j$. |
|
1995 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1996 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1997 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1998 |
of as $n$-category $k{-}1$-sphere modules |
|
1999 |
(generalizations of bimodules). |
|
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2000 |
On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2001 |
and from this point of view it is clear that they satisfy all of the axioms of an |
208 | 2002 |
$n{+}1$-category. |
2003 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
2004 |
||
2005 |
\medskip |
|
2006 |
||
2007 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2008 |
The construction of the 0- through $n$-morphisms was easy and tautological, but the |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2009 |
$n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2010 |
duality assumptions on the lower morphisms. These are required because we define the spaces of $n{+}1$-morphisms by making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. The additional duality assumptions are needed to prove independence of our definition form these choices. |
208 | 2011 |
|
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2012 |
Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2013 |
by a cell complex labeled by 0- through $n$-morphisms, as above. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2014 |
Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2015 |
$\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2016 |
Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2017 |
Recall from above the associated 1-category $\cS(E_c)$. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2018 |
We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2019 |
Define |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2020 |
\[ |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2021 |
\cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2022 |
\] |
208 | 2023 |
|
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2024 |
We will show that if the sphere modules are equipped with a `compatible family of |
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2025 |
non-degenerate inner products', then there is a coherent family of isomorphisms |
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2026 |
$\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2027 |
This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
208 | 2028 |
|
390
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2029 |
First we must define ``inner product", ``non-degenerate" and ``compatible". |
387
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2030 |
Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2031 |
(We assume we are working in the unoriented category.) |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2032 |
Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2033 |
along their common boundary. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2034 |
An {\it inner product} on $\cS(Y)$ is a dual vector |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2035 |
\[ |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2036 |
z_Y : \cS(Y\cup\ol{Y}) \to \c. |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2037 |
\] |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2038 |
We will also use the notation |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2039 |
\[ |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2040 |
\langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2041 |
\] |
390
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2042 |
An inner product induces a linear map |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2043 |
\begin{eqnarray*} |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2044 |
\varphi: \cS(Y) &\to& \cS(Y)^* \\ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2045 |
a &\mapsto& \langle a, \cdot \rangle |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2046 |
\end{eqnarray*} |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2047 |
which satisfies, for all morphisms $e$ of $\cS(\bd Y)$, |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2048 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2049 |
\varphi(ae)(b) = \langle ae, b \rangle = z_Y(a\bullet e\bullet b) = |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2050 |
\langle a, eb \rangle = \varphi(a)(eb) . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2051 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2052 |
In other words, $\varphi$ is a map of $\cS(\bd Y)$ modules. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2053 |
An inner product is {\it non-degenerate} if $\varphi$ is an isomorphism. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2054 |
This implies that $\cS(Y; c)$ is finite dimensional for all boundary conditions $c$. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2055 |
(One can think of these inner products as giving some duality in dimension $n{+}1$; |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2056 |
heretofore we have only assumed duality in dimensions 0 through $n$.) |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2057 |
|
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2058 |
Next we define compatibility. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2059 |
Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2060 |
Let $X_1$ and $X_2$ be the two components of $Y\times I$ cut along |
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2061 |
$D\times I$, in both cases using the pinched product. |
390
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2062 |
(Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2063 |
manifold.) |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2064 |
We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
393 | 2065 |
(see Figure \ref{jun23a}). |
2066 |
\begin{figure}[t] |
|
2067 |
\begin{equation*} |
|
2068 |
\mathfig{.6}{tempkw/jun23a} |
|
2069 |
\end{equation*} |
|
2070 |
\caption{$Y\times I$ sliced open} |
|
2071 |
\label{jun23a} |
|
2072 |
\end{figure} |
|
390
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2073 |
Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2074 |
which agree on their boundaries, we can evaluate |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2075 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2076 |
z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2077 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2078 |
(This requires a choice of homeomorphism $Y_i \cup \ol{Y}_i \cup (D\times I) \cong |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2079 |
Y_i \cup \ol{Y}_i$, but the value of $z_{Y_i}$ is independent of this choice.) |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2080 |
We can think of $z_{Y_i}$ as giving a function |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2081 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2082 |
\psi_i : \cS(Y_i) \ot \cS(\ol{Y}_i) \to \cS(D\times I)^* |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2083 |
\stackrel{\varphi\inv}{\longrightarrow} \cS(D\times I) . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2084 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2085 |
We can now finally define a family of inner products to be {\it compatible} if |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2086 |
for all decompositions $Y = Y_1\cup Y_2$ as above and all $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2087 |
we have |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2088 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2089 |
z_Y(a_1\bullet a_2\bullet b_1\bullet b_2) = |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2090 |
z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2091 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2092 |
In other words, the inner product on $Y$ is determined by the inner products on |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2093 |
$Y_1$, $Y_2$ and $D\times I$. |
207 | 2094 |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2095 |
Now we show how to unambiguously identify $\cS(X; c; E)$ and $\cS(X; c; E')$ for any |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2096 |
two choices of $E$ and $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2097 |
Consider first the case where $\bd X$ is decomposed as three $n$-balls $A$, $B$ and $C$, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2098 |
with $E = \bd(A\cup B)$ and $E' = \bd A$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2099 |
We must provide an isomorphism between $\cS(X; c; E) = \hom(\cS(C), \cS(A\cup B))$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2100 |
and $\cS(X; c; E') = \hom(\cS(C\cup \ol{B}), \cS(A))$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2101 |
Let $D = B\cap A$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2102 |
Then as above we can construct a map |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2103 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2104 |
\psi: \cS(B)\ot\cS(\ol{B}) \to \cS(D\times I) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2105 |
\] |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2106 |
Given $f\in \hom(\cS(C), \cS(A\cup B))$ we define $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2107 |
to be the composition |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2108 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2109 |
\cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2110 |
\cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2111 |
\cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2112 |
\] |
393 | 2113 |
(See Figure \ref{jun23b}.) |
2114 |
\begin{figure}[t] |
|
2115 |
\begin{equation*} |
|
2116 |
\mathfig{.5}{tempkw/jun23b} |
|
2117 |
\end{equation*} |
|
2118 |
\caption{Moving $B$ from top to bottom} |
|
2119 |
\label{jun23b} |
|
2120 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2121 |
Let $D' = B\cap C$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2122 |
Using the inner products there is an adjoint map |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2123 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2124 |
\psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2125 |
\] |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2126 |
Given $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ we define $f\in \hom(\cS(C), \cS(A\cup B))$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2127 |
to be the composition |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2128 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2129 |
\cS(C) \stackrel{\cong}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2130 |
\cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2131 |
\cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2132 |
\cS(A\cup B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2133 |
\] |
393 | 2134 |
(See Figure \ref{jun23c}.) |
2135 |
\begin{figure}[t] |
|
2136 |
\begin{equation*} |
|
2137 |
\mathfig{.5}{tempkw/jun23c} |
|
2138 |
\end{equation*} |
|
2139 |
\caption{Moving $B$ from bottom to top} |
|
2140 |
\label{jun23c} |
|
2141 |
\end{figure} |
|
2142 |
Let $D' = B\cap C$. |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2143 |
It is not hard too show that the above two maps are mutually inverse. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2144 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2145 |
\begin{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2146 |
Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2147 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2148 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2149 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2150 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2151 |
$E$ and $E'$ are isotopic, and any isotopy is |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2152 |
homotopic to a composition of small isotopies which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2153 |
(a) supported away from $E$, or (b) modify $E$ in the simple manner described above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2154 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2155 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2156 |
It follows from the lemma that we can construct an isomorphism |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2157 |
between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2158 |
This construction involves on a choice of simple ``moves" (as above) to transform |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2159 |
$E$ to $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2160 |
We must now show that the isomorphism does not depend on this choice. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2161 |
We will show below that it suffice to check two ``movie moves". |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2162 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2163 |
The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2164 |
The result is equivalent to doing nothing. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2165 |
As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2166 |
inverse, so we have invariance under this movie move. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2167 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2168 |
The second movie move replaces to successive pushes in the same direction, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2169 |
across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
393 | 2170 |
(See Figure \ref{jun23d}.) |
2171 |
\begin{figure}[t] |
|
2172 |
\begin{equation*} |
|
2173 |
\mathfig{.9}{tempkw/jun23d} |
|
2174 |
\end{equation*} |
|
2175 |
\caption{A movie move} |
|
2176 |
\label{jun23d} |
|
2177 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2178 |
Invariance under this movie move follows from the compatibility of the inner |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2179 |
product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
411
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
410
diff
changeset
|
2180 |
\nn{should also say something about locality/distant-commutativity} |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2181 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2182 |
If $n\ge 2$, these two movie move suffice: |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2183 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2184 |
\begin{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2185 |
Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2186 |
The any two sequences of elementary moves connecting $E$ to $E'$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2187 |
are related by a sequence of the two movie moves defined above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2188 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2189 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2190 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2191 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2192 |
Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2193 |
of $\bd X$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2194 |
Up to homotopy, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2195 |
such a family is homotopic to a family which can be decomposed |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2196 |
into small families which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2197 |
(a) supported away from $E$, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2198 |
(b) have boundaries corresponding to the two movie moves above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2199 |
Finally, observe that the space of $E$'s is simply connected. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2200 |
(This fails for $n=1$.) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2201 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2202 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2203 |
For $n=1$ we have to check an additional ``global" relations corresponding to |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2204 |
rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2205 |
\nn{should check this global move, or maybe cite Frobenius reciprocity result} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2206 |
|
207 | 2207 |
\nn{...} |
101 | 2208 |
|
2209 |
\medskip |
|
2210 |
\hrule |
|
2211 |
\medskip |
|
2212 |
||
95 | 2213 |
\nn{to be continued...} |
101 | 2214 |
\medskip |
98 | 2215 |
|
2216 |
||
208 | 2217 |
|
2218 |
||
2219 |
||
2220 |
||
98 | 2221 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2222 |
a separate paper): |
|
2223 |
\begin{itemize} |
|
207 | 2224 |
\item discuss Morita equivalence |
139 | 2225 |
\item functors |
98 | 2226 |
\end{itemize} |
2227 |
||
204 | 2228 |