author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Tue, 16 Mar 2010 14:11:07 +0000 | |
changeset 220 | d31a9c505f29 |
parent 218 | 1acb5f508cf6 |
child 222 | 217b6a870532 |
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 |
|
141 | 6 |
\section{$n$-categories} |
94 | 7 |
\label{sec:ncats} |
8 |
||
108 | 9 |
\subsection{Definition of $n$-categories} |
10 |
||
94 | 11 |
Before proceeding, we need more appropriate definitions of $n$-categories, |
12 |
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
|
187 | 13 |
(As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of |
14 |
a `weak' $n$-category with `strong duality'.) |
|
94 | 15 |
|
141 | 16 |
The definitions presented below tie the categories more closely to the topology |
17 |
and avoid combinatorial questions about, for example, the minimal sufficient |
|
18 |
collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
|
19 |
For examples of topological origin, it is typically easy to show that they |
|
20 |
satisfy our axioms. |
|
21 |
For examples of a more purely algebraic origin, one would typically need the combinatorial |
|
22 |
results that we have avoided here. |
|
23 |
||
24 |
\medskip |
|
25 |
||
94 | 26 |
Consider first ordinary $n$-categories. |
195 | 27 |
\nn{Actually, we're doing both plain and infinity cases here} |
94 | 28 |
We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
29 |
We must decide on the ``shape" of the $k$-morphisms. |
|
30 |
Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
|
31 |
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
|
32 |
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
|
33 |
and so on. |
|
34 |
(This allows for strict associativity.) |
|
145 | 35 |
Still other definitions \nn{need refs for all these; maybe the Leinster book \cite{MR2094071}} |
94 | 36 |
model the $k$-morphisms on more complicated combinatorial polyhedra. |
37 |
||
108 | 38 |
We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
39 |
the standard $k$-ball. |
|
94 | 40 |
In other words, |
41 |
||
187 | 42 |
\begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
43 |
For any $k$-manifold $X$ homeomorphic |
|
103 | 44 |
to the standard $k$-ball, we have a set of $k$-morphisms |
187 | 45 |
$\cC_k(X)$. |
46 |
\end{preliminary-axiom} |
|
94 | 47 |
|
103 | 48 |
Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
49 |
standard $k$-ball. |
|
50 |
We {\it do not} assume that it is equipped with a |
|
51 |
preferred homeomorphism to the standard $k$-ball. |
|
52 |
The same goes for ``a $k$-sphere" below. |
|
53 |
||
109 | 54 |
|
55 |
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
|
56 |
the boundary), we want a corresponding |
|
94 | 57 |
bijection of sets $f:\cC(X)\to \cC(Y)$. |
109 | 58 |
(This will imply ``strong duality", among other things.) |
94 | 59 |
So we replace the above with |
60 |
||
187 | 61 |
\begin{axiom}[Morphisms] |
62 |
\label{axiom:morphisms} |
|
63 |
For each $0 \le k \le n$, we have a functor $\cC_k$ from |
|
103 | 64 |
the category of $k$-balls and |
187 | 65 |
homeomorphisms to the category of sets and bijections. |
66 |
\end{axiom} |
|
67 |
||
94 | 68 |
|
69 |
(Note: We usually omit the subscript $k$.) |
|
70 |
||
195 | 71 |
We are so far being deliberately vague about what flavor of $k$-balls |
72 |
we are considering. |
|
94 | 73 |
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
74 |
They could be topological or PL or smooth. |
|
195 | 75 |
%\nn{need to check whether this makes much difference} |
94 | 76 |
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
77 |
to be fussier about corners.) |
|
78 |
For each flavor of manifold there is a corresponding flavor of $n$-category. |
|
195 | 79 |
We will concentrate on the case of PL unoriented manifolds. |
94 | 80 |
|
81 |
Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
|
82 |
of morphisms). |
|
83 |
The 0-sphere is unusual among spheres in that it is disconnected. |
|
84 |
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
|
109 | 85 |
(Actually, this is only true in the oriented case, with 1-morphsims parameterized |
86 |
by oriented 1-balls.) |
|
94 | 87 |
For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
88 |
\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
|
89 |
We prefer to combine the domain and range into a single entity which we call the |
|
90 |
boundary of a morphism. |
|
91 |
Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
|
92 |
||
205 | 93 |
\nn{perhaps it's better to define $\cC(S^k)$ as a colimit, rather than making it new data} |
94 |
||
187 | 95 |
\begin{axiom}[Boundaries (spheres)] |
96 |
For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
|
103 | 97 |
the category of $k$-spheres and |
187 | 98 |
homeomorphisms to the category of sets and bijections. |
99 |
\end{axiom} |
|
94 | 100 |
|
101 |
(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
|
102 |
||
195 | 103 |
\begin{axiom}[Boundaries (maps)]\label{nca-boundary} |
187 | 104 |
For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
105 |
These maps, for various $X$, comprise a natural transformation of functors. |
|
106 |
\end{axiom} |
|
94 | 107 |
|
108 |
(Note that the first ``$\bd$" above is part of the data for the category, |
|
109 |
while the second is the ordinary boundary of manifolds.) |
|
110 |
||
109 | 111 |
Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
94 | 112 |
|
113 |
Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
|
103 | 114 |
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
94 | 115 |
all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
116 |
(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
|
117 |
and all the structure maps of the $n$-category should be compatible with the auxiliary |
|
118 |
category structure. |
|
119 |
Note that this auxiliary structure is only in dimension $n$; |
|
120 |
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
|
121 |
||
122 |
\medskip |
|
109 | 123 |
\nn{ |
124 |
%At the moment I'm a little confused about orientations, and more specifically |
|
125 |
%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
|
126 |
Maybe need a discussion about what the boundary of a manifold with a |
|
127 |
structure (e.g. orientation) means. |
|
94 | 128 |
Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
129 |
Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
|
130 |
first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
|
131 |
equipped with an orientation of its once-stabilized tangent bundle. |
|
132 |
Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
|
133 |
their $k$ times stabilized tangent bundles. |
|
141 | 134 |
(cf. [Stolz and Teichner].) |
115 | 135 |
Probably should also have a framing of the stabilized dimensions in order to indicate which |
136 |
side the bounded manifold is on. |
|
94 | 137 |
For the moment just stick with unoriented manifolds.} |
138 |
\medskip |
|
139 |
||
140 |
We have just argued that the boundary of a morphism has no preferred splitting into |
|
141 |
domain and range, but the converse meets with our approval. |
|
142 |
That is, given compatible domain and range, we should be able to combine them into |
|
143 |
the full boundary of a morphism: |
|
144 |
||
187 | 145 |
\begin{axiom}[Boundary from domain and range] |
146 |
Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$, |
|
179 | 147 |
$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
94 | 148 |
Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
149 |
two maps $\bd: \cC(B_i)\to \cC(E)$. |
|
187 | 150 |
Then we have an injective map |
94 | 151 |
\[ |
152 |
\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
|
153 |
\] |
|
187 | 154 |
which is natural with respect to the actions of homeomorphisms. |
155 |
\end{axiom} |
|
94 | 156 |
|
179 | 157 |
\begin{figure}[!ht] |
186 | 158 |
$$ |
159 |
\begin{tikzpicture}[every label/.style={green}] |
|
160 |
\node[fill=black, circle, label=below:$E$](S) at (0,0) {}; |
|
161 |
\node[fill=black, circle, label=above:$E$](N) at (0,2) {}; |
|
162 |
\draw (S) arc (-90:90:1); |
|
163 |
\draw (N) arc (90:270:1); |
|
164 |
\node[left] at (-1,1) {$B_1$}; |
|
165 |
\node[right] at (1,1) {$B_2$}; |
|
166 |
\end{tikzpicture} |
|
167 |
$$ |
|
179 | 168 |
$$\mathfig{.4}{tempkw/blah3}$$ |
195 | 169 |
\caption{Combining two balls to get a full boundary |
170 |
\nn{maybe smaller dots for $E$ in pdf fig}}\label{blah3}\end{figure} |
|
179 | 171 |
|
94 | 172 |
Note that we insist on injectivity above. |
109 | 173 |
|
94 | 174 |
Let $\cC(S)_E$ denote the image of $\gl_E$. |
109 | 175 |
We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
176 |
||
195 | 177 |
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
178 |
as above, then we define $\cC(X)_E = \bd^{-1}(\cC(\bd X)_E)$. |
|
179 |
||
110 | 180 |
We will call the projection $\cC(S)_E \to \cC(B_i)$ |
181 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
|
182 |
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
|
195 | 183 |
More generally, we also include under the rubric ``restriction map" the |
184 |
the boundary maps of Axiom \ref{nca-boundary} above, |
|
185 |
another calss of maps introduced after Axion \ref{nca-assoc} below, as well as any composition |
|
186 |
of restriction maps (inductive definition). |
|
187 |
In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
|
188 |
($i = 1, 2$, notation from previous paragraph). |
|
189 |
These restriction maps can be thought of as |
|
190 |
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
|
94 | 191 |
|
192 |
||
193 |
Next we consider composition of morphisms. |
|
194 |
For $n$-categories which lack strong duality, one usually considers |
|
195 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
196 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
197 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
198 |
one general type of composition which can be in any ``direction". |
|
199 |
||
187 | 200 |
\begin{axiom}[Composition] |
201 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
|
179 | 202 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 203 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 204 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
205 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
206 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
207 |
Then (axiom) we have a map |
|
208 |
\[ |
|
209 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
210 |
\] |
|
211 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
212 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
213 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
187 | 214 |
(For $k=n$, see below.) |
215 |
\end{axiom} |
|
94 | 216 |
|
179 | 217 |
\begin{figure}[!ht] |
218 |
$$\mathfig{.4}{tempkw/blah5}$$ |
|
219 |
\caption{From two balls to one ball}\label{blah5}\end{figure} |
|
220 |
||
195 | 221 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 222 |
The composition (gluing) maps above are strictly associative. |
223 |
\end{axiom} |
|
102 | 224 |
|
179 | 225 |
\begin{figure}[!ht] |
226 |
$$\mathfig{.65}{tempkw/blah6}$$ |
|
227 |
\caption{An example of strict associativity}\label{blah6}\end{figure} |
|
228 |
||
192 | 229 |
\nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
230 |
||
109 | 231 |
Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
110 | 232 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
195 | 233 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
234 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
|
235 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
236 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 237 |
|
192 | 238 |
We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
239 |
We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
|
240 |
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
|
109 | 241 |
|
193 | 242 |
More generally, let $\alpha$ be a subdivision of a ball (or sphere) $X$ into smaller balls. |
243 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
|
244 |
the smaller balls to $X$. |
|
245 |
We will also say that $\cC(X)_\alpha$ are morphisms which are splittable along $\alpha$. |
|
246 |
In situations where the subdivision is notationally anonymous, we will write |
|
247 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
|
248 |
the unnamed subdivision. |
|
249 |
If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$; |
|
250 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
|
251 |
subdivision of $\bd X$ and no competing subdivision of $X$. |
|
192 | 252 |
|
253 |
The above two composition axioms are equivalent to the following one, |
|
102 | 254 |
which we state in slightly vague form. |
255 |
||
256 |
\xxpar{Multi-composition:} |
|
257 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
258 |
into small $k$-balls, there is a |
|
259 |
map from an appropriate subset (like a fibered product) |
|
193 | 260 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 261 |
and these various $m$-fold composition maps satisfy an |
179 | 262 |
operad-type strict associativity condition (Figure \ref{blah7}).} |
263 |
||
264 |
\begin{figure}[!ht] |
|
265 |
$$\mathfig{.8}{tempkw/blah7}$$ |
|
266 |
\caption{Operadish composition and associativity}\label{blah7}\end{figure} |
|
95 | 267 |
|
268 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
269 |
||
187 | 270 |
\begin{axiom}[Product (identity) morphisms] |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
271 |
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
272 |
\begin{enumerate} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
273 |
\item |
95 | 274 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
275 |
\[ \xymatrix{ |
|
96 | 276 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
95 | 277 |
X \ar[r]^{f} & X' |
278 |
} \] |
|
109 | 279 |
commutes, then we have |
280 |
\[ |
|
281 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
282 |
\] |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
283 |
\item |
109 | 284 |
Product morphisms are compatible with gluing (composition) in both factors: |
285 |
\[ |
|
286 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
287 |
\] |
|
288 |
and |
|
289 |
\[ |
|
290 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
291 |
\] |
|
122 | 292 |
\nn{if pinched boundary, then remove first case above} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
293 |
\item |
109 | 294 |
Product morphisms are associative: |
295 |
\[ |
|
296 |
(a\times D)\times D' = a\times (D\times D') . |
|
297 |
\] |
|
298 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
299 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
300 |
\item |
110 | 301 |
Product morphisms are compatible with restriction: |
302 |
\[ |
|
303 |
\res_{X\times E}(a\times D) = a\times E |
|
304 |
\] |
|
305 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
306 |
\end{enumerate} |
187 | 307 |
\end{axiom} |
95 | 308 |
|
110 | 309 |
\nn{need even more subaxioms for product morphisms?} |
95 | 310 |
|
122 | 311 |
\nn{Almost certainly we need a little more than the above axiom. |
312 |
More specifically, in order to bootstrap our way from the top dimension |
|
313 |
properties of identity morphisms to low dimensions, we need regular products, |
|
314 |
pinched products and even half-pinched products. |
|
142 | 315 |
I'm not sure what the best way to cleanly axiomatize the properties of these various |
316 |
products is. |
|
122 | 317 |
For the moment, I'll assume that all flavors of the product are at |
318 |
our disposal, and I'll plan on revising the axioms later.} |
|
319 |
||
128 | 320 |
\nn{current idea for fixing this: make the above axiom a ``preliminary version" |
321 |
(as we have already done with some of the other axioms), then state the official |
|
322 |
axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
|
323 |
one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
|
324 |
the alternative is to give some sort of local criterion for what's allowed. |
|
325 |
state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
|
326 |
} |
|
327 |
||
95 | 328 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
329 |
The last axiom (below), concerning actions of |
|
330 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
331 |
||
332 |
We start with the plain $n$-category case. |
|
333 |
||
187 | 334 |
\begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$} |
335 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
95 | 336 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
187 | 337 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
338 |
\end{preliminary-axiom} |
|
96 | 339 |
|
174 | 340 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 341 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 342 |
Let $J$ be a 1-ball (interval). |
343 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 344 |
(Here we use the ``pinched" version of $Y\times J$. |
345 |
\nn{need notation for this}) |
|
96 | 346 |
We define a map |
347 |
\begin{eqnarray*} |
|
348 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
349 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
350 |
\end{eqnarray*} |
|
142 | 351 |
(See Figure \ref{glue-collar}.) |
189 | 352 |
\begin{figure}[!ht] |
353 |
\begin{equation*} |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
354 |
\begin{tikzpicture} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
355 |
\def\rad{1} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
356 |
\def\srad{0.75} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
357 |
\def\gap{4.5} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
358 |
\foreach \i in {0, 1, 2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
359 |
\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
|
360 |
\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
|
361 |
\foreach \n in {1,2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
362 |
\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
|
363 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
364 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
365 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
366 |
\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
367 |
\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
|
368 |
\end{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
369 |
\node[right=1mm] at (0.east) {$a$}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
370 |
\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
|
371 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
372 |
\draw (1-small) circle (\srad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
373 |
\foreach \theta in {90, 72, ..., -90} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
374 |
\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
375 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
376 |
\filldraw[fill=white] (1) circle (\rad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
377 |
\foreach \n in {1,2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
378 |
\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
|
379 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
380 |
\node[below] at (1-small.south) {$a \times J$}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
381 |
\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
|
382 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
383 |
\begin{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
384 |
\path[clip] (2) circle (\rad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
385 |
\draw[clip] (2.east) circle (\srad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
386 |
\foreach \y in {1, 0.86, ..., -1} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
387 |
\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
388 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
389 |
\end{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
390 |
\end{tikzpicture} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
391 |
\end{equation*} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
392 |
\begin{equation*} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
393 |
\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 | 394 |
\end{equation*} |
395 |
||
396 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
174 | 397 |
We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
398 |
\nn{bad terminology; fix it later} |
|
399 |
\nn{also need to make clear that plain old isotopic to the identity implies |
|
400 |
extended isotopic} |
|
97 | 401 |
\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
402 |
extended isotopies are also plain isotopies, so |
|
403 |
no extension necessary} |
|
96 | 404 |
It can be thought of as the action of the inverse of |
405 |
a map which projects a collar neighborhood of $Y$ onto $Y$. |
|
406 |
||
407 |
The revised axiom is |
|
408 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
409 |
\stepcounter{axiom} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
410 |
\begin{axiom-numbered}{\arabic{axiom}a}{Extended isotopy invariance in dimension $n$} |
187 | 411 |
\label{axiom:extended-isotopies} |
412 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
174 | 413 |
to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
187 | 414 |
Then $f$ acts trivially on $\cC(X)$. |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
415 |
\end{axiom-numbered} |
96 | 416 |
|
417 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
94 | 418 |
|
97 | 419 |
\smallskip |
420 |
||
421 |
For $A_\infty$ $n$-categories, we replace |
|
422 |
isotopy invariance with the requirement that families of homeomorphisms act. |
|
423 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
|
424 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
425 |
\begin{axiom-numbered}{\arabic{axiom}b}{Families of homeomorphisms act in dimension $n$} |
187 | 426 |
For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
97 | 427 |
\[ |
428 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
429 |
\] |
|
430 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
|
431 |
which fix $\bd X$. |
|
432 |
These action maps are required to be associative up to homotopy |
|
433 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
434 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
435 |
\nn{repeat diagram here?} |
|
187 | 436 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
437 |
\end{axiom-numbered} |
97 | 438 |
|
439 |
We should strengthen the above axiom to apply to families of extended homeomorphisms. |
|
109 | 440 |
To do this we need to explain how extended homeomorphisms form a topological space. |
97 | 441 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
442 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
|
443 |
\nn{need to also say something about collaring homeomorphisms.} |
|
444 |
\nn{this paragraph needs work.} |
|
445 |
||
103 | 446 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
447 |
into a plain $n$-category (enriched over graded groups). |
|
97 | 448 |
\nn{say more here?} |
449 |
In the other direction, if we enrich over topological spaces instead of chain complexes, |
|
450 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
|
451 |
instead of $C_*(\Homeo_\bd(X))$. |
|
452 |
Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
|
453 |
type $A_\infty$ $n$-category. |
|
454 |
||
99 | 455 |
\medskip |
97 | 456 |
|
99 | 457 |
The alert reader will have already noticed that our definition of (plain) $n$-category |
458 |
is extremely similar to our definition of topological fields. |
|
142 | 459 |
The main difference is that for the $n$-category definition we restrict our attention to balls |
99 | 460 |
(and their boundaries), while for fields we consider all manifolds. |
142 | 461 |
(A minor difference is that in the category definition we directly impose isotopy |
462 |
invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields |
|
463 |
but then mod out by local relations which imply isotopy invariance.) |
|
99 | 464 |
Thus a system of fields determines an $n$-category simply by restricting our attention to |
465 |
balls. |
|
142 | 466 |
This $n$-category can be thought of as the local part of the fields. |
99 | 467 |
Conversely, given an $n$-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
|
468 |
a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 469 |
|
142 | 470 |
%\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
471 |
%of fields. |
|
472 |
%The universal (colimit) construction becomes our generalized definition of blob homology. |
|
473 |
%Need to explain how it relates to the old definition.} |
|
97 | 474 |
|
95 | 475 |
\medskip |
476 |
||
195 | 477 |
\subsection{Examples of $n$-categories}\ \ |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
478 |
|
101 | 479 |
\nn{these examples need to be fleshed out a bit more} |
480 |
||
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
481 |
We now describe several classes of examples of $n$-categories satisfying our axioms. |
101 | 482 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
483 |
\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
|
484 |
\rm |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
485 |
\label{ex: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
|
486 |
Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
487 |
For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
488 |
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
|
489 |
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 490 |
homotopies fixed on $\bd X$. |
101 | 491 |
(Note that homotopy invariance implies isotopy invariance.) |
492 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
493 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
494 |
\end{example} |
101 | 495 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
496 |
\begin{example}[Maps to a space, with a fiber] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
497 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
498 |
\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 499 |
We can modify the example above, by fixing a |
500 |
closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. 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
|
501 |
\end{example} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
502 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
503 |
\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
|
504 |
\rm |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
505 |
\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
|
506 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 507 |
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
|
508 |
(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
|
509 |
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
|
510 |
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
|
511 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 512 |
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
|
513 |
$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
101 | 514 |
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
515 |
\end{example} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
516 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
517 |
The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
518 |
\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
|
519 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
520 |
\label{ex:traditional-n-categories} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
521 |
Given a `traditional $n$-category with strong duality' $C$ |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
522 |
define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$, |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
523 |
to be the set of all $C$-labeled sub cell complexes of $X$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
524 |
(See Subsection \ref{sec:fields}.) |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
525 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
526 |
combinations of $C$-labeled sub cell complexes of $X$ |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
527 |
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
|
528 |
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
529 |
with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
530 |
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
|
531 |
Define $\cC(X)$, for $\dim(X) < n$, |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
532 |
to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
533 |
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
|
534 |
to be the dual Hilbert space $A(X\times F; c)$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
535 |
\nn{refer elsewhere for details?} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
536 |
\end{example} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
537 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
538 |
Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
204 | 539 |
|
540 |
\nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
|
541 |
||
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
542 |
\newcommand{\Bord}{\operatorname{Bord}} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
543 |
\begin{example}[The bordism $n$-category] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
544 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
545 |
\label{ex:bordism-category} |
196 | 546 |
For a $k$-ball or $k$-sphere $X$, $k<n$, 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
|
547 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
196 | 548 |
to $\bd X$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
549 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
550 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 551 |
$W \to W'$ 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
|
552 |
\end{example} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
553 |
|
196 | 554 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 555 |
|
196 | 556 |
%\begin{example}[Variation on the above examples] |
557 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
558 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
559 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
560 |
%%to think of these guys as affording a representation |
|
561 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
562 |
%\end{example} |
|
101 | 563 |
|
564 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
565 |
We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 566 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
567 |
\begin{example}[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
|
568 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
569 |
\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
|
570 |
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 $T$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
571 |
For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps{X \to T}$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
572 |
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 |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
573 |
$$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 574 |
and $C_*$ denotes singular chains. |
211 | 575 |
\nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
576 |
\end{example} |
101 | 577 |
|
211 | 578 |
See \ref{thm:map-recon} below, recovering $C_*(\Maps{M \to T})$ as (up to homotopy) the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
579 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
580 |
\begin{example}[Blob complexes of balls (with a fiber)] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
581 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
582 |
\label{ex:blob-complexes-of-balls} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
583 |
Fix an $m$-dimensional manifold $F$. |
101 | 584 |
Given a plain $n$-category $C$, |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
585 |
when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
586 |
define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
587 |
where $\bc^C_*$ denotes the blob complex based on $C$. |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
588 |
\end{example} |
101 | 589 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
590 |
This example will be essential for ???, which relates ... |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
591 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
592 |
\begin{example} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
593 |
\nn{should add $\infty$ version of bordism $n$-cat} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
594 |
\end{example} |
95 | 595 |
|
108 | 596 |
|
597 |
||
598 |
||
599 |
||
600 |
||
601 |
\subsection{From $n$-categories to systems of fields} |
|
113 | 602 |
\label{ss:ncat_fields} |
197 | 603 |
In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds. |
204 | 604 |
In the case of plain $n$-categories, this is just the usual construction of a TQFT |
605 |
from an $n$-category. |
|
606 |
For $\infty$ $n$-categories \nn{or whatever we decide to call them}, this gives an alternate (and |
|
607 |
somewhat more canonical/tautological) construction of the blob complex. |
|
608 |
\nn{though from this point of view it seems more natural to just add some |
|
609 |
adjective to ``TQFT" rather than coining a completely new term like ``blob complex".} |
|
108 | 610 |
|
197 | 611 |
We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
612 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
|
613 |
We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting system of fields 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 | 614 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
615 |
\begin{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
616 |
Say that a `permissible decomposition' of $W$ is a cell decomposition |
108 | 617 |
\[ |
618 |
W = \bigcup_a X_a , |
|
619 |
\] |
|
142 | 620 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
621 |
|
108 | 622 |
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
|
623 |
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
|
624 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
625 |
The category $\cJ(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
626 |
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
|
627 |
\end{defn} |
119 | 628 |
|
629 |
\begin{figure}[!ht] |
|
630 |
\begin{equation*} |
|
631 |
\mathfig{.63}{tempkw/zz2} |
|
632 |
\end{equation*} |
|
633 |
\caption{A small part of $\cJ(W)$} |
|
634 |
\label{partofJfig} |
|
635 |
\end{figure} |
|
636 |
||
108 | 637 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
638 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
639 |
An $n$-category $\cC$ determines |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
640 |
a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets |
108 | 641 |
(possibly with additional structure if $k=n$). |
197 | 642 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
643 |
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
|
644 |
are splittable along this decomposition. |
|
645 |
%For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell. |
|
108 | 646 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
647 |
\begin{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
648 |
Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
649 |
For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
650 |
\begin{equation} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
651 |
\label{eq:psi-C} |
197 | 652 |
\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
|
653 |
\end{equation} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
654 |
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
|
655 |
$X_a$ 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
|
656 |
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
|
657 |
\end{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
658 |
|
197 | 659 |
When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is a |
660 |
closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
661 |
we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
197 | 662 |
Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
663 |
fix a field on $\bd W$ |
|
664 |
(i.e. fix an element of the colimit associated to $\bd W$). |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
665 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
666 |
Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
667 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
668 |
\begin{defn}[System of fields functor] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
669 |
If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
670 |
That is, for each decomposition $x$ there is a map |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
671 |
$\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
108 | 672 |
above, and $\cC(W)$ 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
|
673 |
\end{defn} |
112 | 674 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
675 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
676 |
When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ is defined as above, as the colimit of $\psi_{\cC;W}$. When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
677 |
\end{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
678 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
679 |
We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
111 | 680 |
|
197 | 681 |
We now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
682 |
\begin{equation*} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
683 |
\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
684 |
\end{equation*} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
685 |
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
|
686 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
687 |
\to \psi_{\cC;W,c}(y)$ 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
|
688 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
689 |
In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 690 |
is more involved. |
142 | 691 |
%\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
|
692 |
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
693 |
Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
694 |
Define $V$ as a vector space via |
112 | 695 |
\[ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
696 |
V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 697 |
\] |
198 | 698 |
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
699 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 700 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
701 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
702 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
|
703 |
\[ |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
704 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 705 |
\] |
706 |
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 | 707 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
112 | 708 |
\nn{need to say this better} |
709 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
710 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
711 |
of $A_\infty$ category} |
|
108 | 712 |
|
113 | 713 |
We will call $m$ the filtration degree of the complex. |
714 |
We can think of this construction as starting with a disjoint copy of a complex for each |
|
715 |
permissible decomposition (filtration degree 0). |
|
716 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
717 |
(filtration degree 1). |
|
198 | 718 |
Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2). |
113 | 719 |
And so on. |
720 |
||
108 | 721 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
722 |
||
723 |
It is easy to see that |
|
724 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
|
725 |
comprise a natural transformation of functors. |
|
726 |
||
727 |
\nn{need to finish explaining why we have a system of fields; |
|
728 |
need to say more about ``homological" fields? |
|
729 |
(actions of homeomorphisms); |
|
730 |
define $k$-cat $\cC(\cdot\times W)$} |
|
731 |
||
732 |
||
733 |
||
734 |
\subsection{Modules} |
|
95 | 735 |
|
101 | 736 |
Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
737 |
a.k.a.\ actions). |
|
199 | 738 |
The definition will be very similar to that of $n$-categories, |
739 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
|
109 | 740 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
198 | 741 |
%\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
742 |
||
104 | 743 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 744 |
in the context of an $m{+}1$-dimensional TQFT. |
745 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
746 |
This will be explained in more detail as we present the axioms. |
|
747 |
||
748 |
Fix an $n$-category $\cC$. |
|
749 |
||
750 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
751 |
(standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
|
752 |
We call $B$ the ball and $N$ the marking. |
|
753 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
754 |
restricts to a homeomorphism of markings. |
|
755 |
||
199 | 756 |
\mmpar{Module axiom 1}{Module morphisms} |
102 | 757 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
758 |
the category of marked $k$-balls and |
|
759 |
homeomorphisms to the category of sets and bijections.} |
|
760 |
||
761 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
762 |
||
104 | 763 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
764 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
765 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
766 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
767 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
768 |
(The union is along $N\times \bd W$.) |
|
110 | 769 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
770 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 771 |
|
182 | 772 |
\begin{figure}[!ht] |
773 |
$$\mathfig{.8}{tempkw/blah15}$$ |
|
774 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
|
775 |
||
103 | 776 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
777 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 778 |
|
199 | 779 |
\mmpar{Module axiom 2}{Module boundaries (hemispheres)} |
102 | 780 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
104 | 781 |
the category of marked $k$-hemispheres and |
102 | 782 |
homeomorphisms to the category of sets and bijections.} |
783 |
||
104 | 784 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
785 |
||
199 | 786 |
\mmpar{Module axiom 3}{Module boundaries (maps)} |
102 | 787 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
788 |
These maps, for various $M$, comprise a natural transformation of functors.} |
|
789 |
||
110 | 790 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 791 |
|
792 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
793 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
794 |
and $c\in \cC(\bd M)$. |
|
795 |
||
199 | 796 |
\mmpar{Module axiom 4}{Boundary from domain and range} |
102 | 797 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
104 | 798 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
799 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
|
800 |
two maps $\bd: \cM(M_i)\to \cM(E)$. |
|
102 | 801 |
Then (axiom) we have an injective map |
802 |
\[ |
|
199 | 803 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
102 | 804 |
\] |
805 |
which is natural with respect to the actions of homeomorphisms.} |
|
806 |
||
110 | 807 |
Let $\cM(H)_E$ denote the image of $\gl_E$. |
808 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
|
809 |
||
810 |
||
199 | 811 |
\mmpar{Module axiom 5}{Module to category restrictions} |
103 | 812 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
813 |
$\cM(H)\to \cC(H)$. |
|
814 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
|
815 |
These maps comprise a natural transformation of functors.} |
|
102 | 816 |
|
103 | 817 |
Note that combining the various boundary and restriction maps above |
110 | 818 |
(for both modules and $n$-categories) |
103 | 819 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
820 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 821 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
822 |
cutting submanifolds). |
|
103 | 823 |
This fact will be used below. |
102 | 824 |
|
104 | 825 |
In our example, the various restriction and gluing maps above come from |
826 |
restricting and gluing maps into $T$. |
|
827 |
||
828 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 829 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 830 |
(See Figure \ref{zzz3}.) |
103 | 831 |
|
119 | 832 |
\begin{figure}[!ht] |
833 |
\begin{equation*} |
|
834 |
\mathfig{.63}{tempkw/zz3} |
|
835 |
\end{equation*} |
|
836 |
\caption{Module composition (top); $n$-category action (bottom)} |
|
837 |
\label{zzz3} |
|
838 |
\end{figure} |
|
839 |
||
840 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 841 |
|
200 | 842 |
\mmpar{Module axiom 6}{Module composition} |
103 | 843 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
844 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
|
845 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
846 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
847 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
848 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
849 |
Then (axiom) we have a map |
|
850 |
\[ |
|
851 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
852 |
\] |
|
853 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
854 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
855 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
856 |
(For $k=n$, see below.)} |
|
857 |
||
119 | 858 |
|
859 |
||
103 | 860 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
861 |
module morphism. |
|
862 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
863 |
||
200 | 864 |
\mmpar{Module axiom 7}{$n$-category action} |
103 | 865 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
866 |
$X$ is a plain $k$-ball, |
|
867 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
868 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
869 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
870 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
871 |
Then (axiom) we have a map |
|
872 |
\[ |
|
873 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
874 |
\] |
|
875 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
876 |
to the intersection of the boundaries of $X$ and $M'$. |
|
877 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
878 |
(For $k=n$, see below.)} |
|
879 |
||
200 | 880 |
\mmpar{Module axiom 8}{Strict associativity} |
103 | 881 |
{The composition and action maps above are strictly associative.} |
882 |
||
110 | 883 |
Note that the above associativity axiom applies to mixtures of module composition, |
884 |
action maps and $n$-category composition. |
|
119 | 885 |
See Figure \ref{zzz1b}. |
886 |
||
887 |
\begin{figure}[!ht] |
|
888 |
\begin{equation*} |
|
889 |
\mathfig{1}{tempkw/zz1b} |
|
890 |
\end{equation*} |
|
891 |
\caption{Two examples of mixed associativity} |
|
892 |
\label{zzz1b} |
|
893 |
\end{figure} |
|
894 |
||
110 | 895 |
|
896 |
The above three axioms are equivalent to the following axiom, |
|
103 | 897 |
which we state in slightly vague form. |
898 |
\nn{need figure for this} |
|
899 |
||
900 |
\xxpar{Module multi-composition:} |
|
901 |
{Given any decomposition |
|
902 |
\[ |
|
903 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
904 |
\] |
|
905 |
of a marked $k$-ball $M$ |
|
906 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
907 |
map from an appropriate subset (like a fibered product) |
|
908 |
of |
|
909 |
\[ |
|
910 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
911 |
\] |
|
912 |
to $\cM(M)$, |
|
913 |
and these various multifold composition maps satisfy an |
|
914 |
operad-type strict associativity condition.} |
|
915 |
||
916 |
(The above operad-like structure is analogous to the swiss cheese operad |
|
146 | 917 |
\cite{MR1718089}.) |
200 | 918 |
%\nn{need to double-check that this is true.} |
103 | 919 |
|
200 | 920 |
\mmpar{Module axiom 9}{Product/identity morphisms} |
103 | 921 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
922 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
|
923 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
|
924 |
\[ \xymatrix{ |
|
925 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
|
926 |
M \ar[r]^{f} & M' |
|
927 |
} \] |
|
928 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
929 |
||
111 | 930 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
103 | 931 |
|
200 | 932 |
\nn{postpone finalizing the above axiom until the n-cat version is finalized} |
110 | 933 |
|
103 | 934 |
There are two alternatives for the next axiom, according whether we are defining |
935 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
936 |
In the plain case we require |
|
937 |
||
200 | 938 |
\mmpar{Module axiom 10a}{Extended isotopy invariance in dimension $n$} |
103 | 939 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
175 | 940 |
to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
103 | 941 |
Then $f$ acts trivially on $\cM(M)$.} |
942 |
||
943 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
944 |
||
945 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
946 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
947 |
on $\bd B \setmin N$. |
|
948 |
||
949 |
For $A_\infty$ modules we require |
|
950 |
||
200 | 951 |
\mmpar{Module axiom 10b}{Families of homeomorphisms act} |
103 | 952 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
953 |
\[ |
|
954 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
955 |
\] |
|
956 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
957 |
which fix $\bd M$. |
|
958 |
These action maps are required to be associative up to homotopy |
|
959 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
960 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
961 |
\nn{repeat diagram here?} |
|
962 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
963 |
||
964 |
\medskip |
|
102 | 965 |
|
104 | 966 |
Note that the above axioms imply that an $n$-category module has the structure |
967 |
of an $n{-}1$-category. |
|
968 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
969 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
970 |
above the non-marked boundary component of $J$. |
|
200 | 971 |
(More specifically, we collapse $X\times P$ to a single point, where |
972 |
$P$ is the non-marked boundary component of $J$.) |
|
973 |
\nn{give figure for this?} |
|
104 | 974 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 975 |
|
105 | 976 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
977 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
978 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
979 |
of marked 1-balls, call them left-marked and right-marked, |
|
980 |
and hence there are two types of modules, call them right modules and left modules. |
|
981 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
982 |
there is no left/right module distinction. |
|
983 |
||
130 | 984 |
\medskip |
985 |
||
986 |
Examples of modules: |
|
987 |
\begin{itemize} |
|
142 | 988 |
\item \nn{examples from TQFTs} |
989 |
\item \nn{for maps to $T$, can restrict to subspaces of $T$;} |
|
130 | 990 |
\end{itemize} |
108 | 991 |
|
992 |
\subsection{Modules as boundary labels} |
|
112 | 993 |
\label{moddecss} |
108 | 994 |
|
995 |
Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
143 | 996 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
997 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
998 |
||
999 |
%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
1000 |
%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
1001 |
%component $\bd_i W$ of $W$. |
|
1002 |
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
108 | 1003 |
|
1004 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
|
1005 |
\nn{give ref} |
|
1006 |
(If $k = n$ and our $k$-categories are enriched, then |
|
1007 |
$\cC(W, \cN)$ will have additional structure; see below.) |
|
1008 |
||
1009 |
Define a permissible decomposition of $W$ to be a decomposition |
|
1010 |
\[ |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1011 |
W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
108 | 1012 |
\] |
1013 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
1014 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
143 | 1015 |
with $M_{ib}\cap Y_i$ being the marking. |
1016 |
(See Figure \ref{mblabel}.) |
|
1017 |
\begin{figure}[!ht]\begin{equation*} |
|
1018 |
\mathfig{.9}{tempkw/mblabel} |
|
1019 |
\end{equation*}\caption{A permissible decomposition of a manifold |
|
146 | 1020 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
108 | 1021 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1022 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
1023 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
1024 |
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
|
1025 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
|
1026 |
||
1027 |
$\cN$ determines |
|
1028 |
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
|
1029 |
(possibly with additional structure if $k=n$). |
|
1030 |
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
|
1031 |
\[ |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1032 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 1033 |
\] |
1034 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
1035 |
$X_a$ and $M_{ib}$ all agree. |
|
1036 |
(Think fibered product.) |
|
1037 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
|
1038 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
1039 |
||
1040 |
Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
|
143 | 1041 |
(Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
1042 |
homotopy colimit.) |
|
108 | 1043 |
|
143 | 1044 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1045 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
1046 |
$D\times Y_i \sub \bd(D\times W)$. |
|
112 | 1047 |
|
143 | 1048 |
It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
1049 |
has the structure of an $n{-}k$-category. |
|
144 | 1050 |
|
1051 |
\medskip |
|
1052 |
||
1053 |
||
1054 |
%\subsection{Tensor products} |
|
108 | 1055 |
|
144 | 1056 |
We will use a simple special case of the above |
1057 |
construction to define tensor products |
|
1058 |
of modules. |
|
1059 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
1060 |
(If $k=1$ and manifolds are oriented, then one should be |
|
1061 |
a left module and the other a right module.) |
|
1062 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
1063 |
Define the tensor product of $\cM_1$ and $\cM_2$ to be the |
|
1064 |
$n{-}1$-category $\cT(J, \cM_1, \cM_2)$, |
|
1065 |
\[ |
|
1066 |
\cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) . |
|
1067 |
\] |
|
1068 |
This of course depends (functorially) |
|
1069 |
on the choice of 1-ball $J$. |
|
105 | 1070 |
|
144 | 1071 |
We will define a more general self tensor product (categorified coend) below. |
1072 |
||
112 | 1073 |
|
144 | 1074 |
|
1075 |
||
1076 |
%\nn{what about self tensor products /coends ?} |
|
105 | 1077 |
|
108 | 1078 |
\nn{maybe ``tensor product" is not the best name?} |
1079 |
||
144 | 1080 |
%\nn{start with (less general) tensor products; maybe change this later} |
106 | 1081 |
|
107 | 1082 |
|
1083 |
||
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1084 |
\subsection{The $n{+}1$-category of sphere modules} |
218 | 1085 |
\label{ssec:spherecat} |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1086 |
|
205 | 1087 |
In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1088 |
whose objects correspond to $n$-categories. |
|
1089 |
This is a version of the familiar algebras-bimodules-intertwiners 2-category. |
|
155 | 1090 |
(Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule, |
1091 |
since a 0-sphere has an obvious bi-ness. |
|
1092 |
This is much less true for higher dimensional spheres, |
|
1093 |
so we prefer the term ``sphere module" for the general case.) |
|
144 | 1094 |
|
205 | 1095 |
The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1096 |
these first. |
|
1097 |
The $n{+}1$-dimensional part of $\cS$ consist of intertwiners |
|
1098 |
(of garden-variety $1$-category modules associated to decorated $n$-balls). |
|
1099 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
|
1100 |
the duality requirements of an $n{+}1$-category, we will have to assume |
|
1101 |
that our $n$-categories and modules have non-degenerate inner products. |
|
1102 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
1103 |
||
1104 |
\medskip |
|
1105 |
||
1106 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
|
1107 |
These will be defined in terms of certain classes of marked balls, very similarly |
|
1108 |
to the definition of $n$-category modules above. |
|
1109 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
1110 |
Because of this similarity, we only sketch the definitions below. |
|
1111 |
||
1112 |
We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. |
|
1113 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
|
1114 |
Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
|
1115 |
$(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. |
|
209 | 1116 |
See Figure \ref{feb21a}. |
205 | 1117 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1118 |
||
209 | 1119 |
\begin{figure}[!ht] |
1120 |
\begin{equation*} |
|
1121 |
\mathfig{.85}{tempkw/feb21a} |
|
1122 |
\end{equation*} |
|
1123 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
|
1124 |
\label{feb21a} |
|
1125 |
\end{figure} |
|
1126 |
||
205 | 1127 |
0-marked balls can be cut into smaller balls in various ways. |
1128 |
These smaller balls could be 0-marked or plain. |
|
1129 |
We can also take the boundary of a 0-marked ball, which is 0-marked sphere. |
|
1130 |
||
1131 |
Fix $n$-categories $\cA$ and $\cB$. |
|
1132 |
These will label the two halves of a 0-marked $k$-ball. |
|
1133 |
The 0-sphere module we define next will depend on $\cA$ and $\cB$ |
|
1134 |
(it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
|
1135 |
||
1136 |
An $n$-category 0-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
|
1137 |
of 0-marked $k$-balls, $1\le k \le n$, |
|
1138 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
|
1139 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
1140 |
Given a decomposition of a 0-marked $k$-ball $X$ into smaller balls $X_i$, we have |
|
1141 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
|
1142 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
1143 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
1144 |
Corresponding to this decomposition we have an action and/or composition map |
|
1145 |
from the product of these various sets into $\cM(X)$. |
|
1146 |
||
1147 |
\medskip |
|
107 | 1148 |
|
206 | 1149 |
Part of the structure of an $n$-cat 0-sphere module is captured my saying it is |
1150 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
|
1151 |
of $\cA$ and $\cB$. |
|
1152 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
1153 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
1154 |
\[ |
|
1155 |
\cD(X) \deq \cM(X\times J) . |
|
1156 |
\] |
|
1157 |
The product is pinched over the boundary of $J$. |
|
1158 |
$\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
|
209 | 1159 |
(see Figure \ref{feb21b}). |
206 | 1160 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 1161 |
|
209 | 1162 |
\begin{figure}[!ht] |
1163 |
\begin{equation*} |
|
1164 |
\mathfig{1}{tempkw/feb21b} |
|
1165 |
\end{equation*} |
|
1166 |
\caption{The pinched product $X\times J$} |
|
1167 |
\label{feb21b} |
|
1168 |
\end{figure} |
|
1169 |
||
206 | 1170 |
More generally, consider an interval with interior marked points, and with the complements |
1171 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
1172 |
by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
|
209 | 1173 |
(See Figure \ref{feb21c}.) |
206 | 1174 |
To this data we can apply to coend construction as in Subsection \ref{moddecss} above |
1175 |
to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category. |
|
1176 |
This amounts to a definition of taking tensor products of bimodules over $n$-categories. |
|
205 | 1177 |
|
209 | 1178 |
\begin{figure}[!ht] |
1179 |
\begin{equation*} |
|
1180 |
\mathfig{1}{tempkw/feb21c} |
|
1181 |
\end{equation*} |
|
1182 |
\caption{Marked and labeled 1-manifolds} |
|
1183 |
\label{feb21c} |
|
1184 |
\end{figure} |
|
1185 |
||
206 | 1186 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1187 |
associated to the marked and labeled circle. |
|
209 | 1188 |
(See Figure \ref{feb21c}.) |
206 | 1189 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1190 |
as the 2-ended tensor product of the two bimodules associated to the two intervals. |
|
1191 |
||
1192 |
\medskip |
|
1193 |
||
1194 |
Next we define $n$-category 1-sphere modules. |
|
1195 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
1196 |
circles (1-spheres) which we just introduced. |
|
1197 |
||
1198 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
1199 |
Fix a marked (and labeled) circle $S$. |
|
209 | 1200 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
207 | 1201 |
\nn{I need to make up my mind whether marked things are always labeled too. |
1202 |
For the time being, let's say they are.} |
|
1203 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
|
1204 |
where $B^j$ is the standard $j$-ball. |
|
1205 |
1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
|
1206 |
smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
|
1207 |
We now proceed as in the above module definitions. |
|
1208 |
||
209 | 1209 |
\begin{figure}[!ht] |
1210 |
\begin{equation*} |
|
1211 |
\mathfig{.4}{tempkw/feb21d} |
|
1212 |
\end{equation*} |
|
1213 |
\caption{Cone on a marked circle} |
|
1214 |
\label{feb21d} |
|
1215 |
\end{figure} |
|
1216 |
||
207 | 1217 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1218 |
\[ |
|
1219 |
\cD(X) \deq \cM(X\times C(S)) . |
|
1220 |
\] |
|
1221 |
The product is pinched over the boundary of $C(S)$. |
|
1222 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
1223 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
1224 |
||
1225 |
More generally, consider a 2-manifold $Y$ |
|
1226 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
1227 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
1228 |
the edges of $K$ are labeled by 0-sphere modules, |
|
1229 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
1230 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
1231 |
If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-manifold |
|
1232 |
associated to the (marked, labeled) boundary of $Y$. |
|
1233 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
1234 |
||
1235 |
\medskip |
|
1236 |
||
1237 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
1238 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 1239 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 1240 |
|
1241 |
\medskip |
|
1242 |
||
1243 |
We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
|
1244 |
Choose some collection of $n$-categories, then choose some collections of bimodules for |
|
1245 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
|
1246 |
possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
|
1247 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
|
1248 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
1249 |
There is a wide range of possibilities. |
|
1250 |
$L_0$ could contain infinitely many $n$-categories or just one. |
|
1251 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
|
1252 |
it could contain several. |
|
208 | 1253 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1254 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
1255 |
||
1256 |
We now define $\cS(X)$, for $X$ of dimension at most $n$, to be the set of all |
|
1257 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
|
1258 |
by elements of $L_j$. |
|
1259 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1260 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1261 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1262 |
of as $n$-category $k{-}1$-sphere modules |
|
1263 |
(generalizations of bimodules). |
|
1264 |
On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, |
|
1265 |
and from this (official) point of view it is clear that they satisfy all of the axioms of an |
|
1266 |
$n{+}1$-category. |
|
1267 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
1268 |
||
1269 |
\medskip |
|
1270 |
||
1271 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
1272 |
||
1273 |
||
1274 |
||
1275 |
||
1276 |
||
1277 |
||
207 | 1278 |
|
1279 |
\nn{...} |
|
101 | 1280 |
|
1281 |
\medskip |
|
1282 |
\hrule |
|
1283 |
\medskip |
|
1284 |
||
95 | 1285 |
\nn{to be continued...} |
101 | 1286 |
\medskip |
98 | 1287 |
|
1288 |
||
208 | 1289 |
|
1290 |
||
1291 |
||
1292 |
||
98 | 1293 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1294 |
a separate paper): |
|
1295 |
\begin{itemize} |
|
1296 |
\item spell out what difference (if any) Top vs PL vs Smooth makes |
|
207 | 1297 |
\item discuss Morita equivalence |
130 | 1298 |
\item morphisms of modules; show that it's adjoint to tensor product |
139 | 1299 |
(need to define dual module for this) |
1300 |
\item functors |
|
98 | 1301 |
\end{itemize} |
1302 |
||
204 | 1303 |
\bigskip |
1304 |
||
1305 |
\hrule |
|
134 | 1306 |
\nn{Some salvaged paragraphs that we might want to work back in:} |
204 | 1307 |
\bigskip |
98 | 1308 |
|
134 | 1309 |
Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) |
1310 |
||
1311 |
The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition |
|
1312 |
\begin{align*} |
|
1313 |
\CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
|
1314 |
\end{align*} |
|
1315 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
|
1316 |
||
1317 |
We now give two motivating examples, as theorems constructing other homological systems of fields, |
|
1318 |
||
1319 |
||
1320 |
\begin{thm} |
|
1321 |
For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as |
|
1322 |
\begin{equation*} |
|
1323 |
\Xi(M) = \CM{M}{X}. |
|
1324 |
\end{equation*} |
|
1325 |
\end{thm} |
|
1326 |
||
1327 |
\begin{thm} |
|
1328 |
Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by |
|
1329 |
\begin{equation*} |
|
1330 |
\cF^{\times F}(M) = \cB_*(M \times F, \cF). |
|
1331 |
\end{equation*} |
|
1332 |
\end{thm} |
|
1333 |
We might suggestively write $\cF^{\times F}$ as $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories. |
|
1334 |
||
1335 |
||
1336 |
In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields. |
|
1337 |
||
1338 |
||
1339 |
\begin{thm} |
|
1340 |
\begin{equation*} |
|
1341 |
\cB_*(M, \Xi) \iso \Xi(M) |
|
1342 |
\end{equation*} |
|
1343 |
\end{thm} |
|
1344 |
||
1345 |
\begin{thm}[Product formula] |
|
1346 |
Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields, |
|
1347 |
there is a quasi-isomorphism |
|
1348 |
\begin{align*} |
|
1349 |
\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) |
|
1350 |
\end{align*} |
|
1351 |
\end{thm} |
|
1352 |
||
1353 |
\begin{question} |
|
1354 |
Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
|
1355 |
\end{question} |
|
1356 |
||
1357 |
\hrule |