author | scott@6e1638ff-ae45-0410-89bd-df963105f760 |
Wed, 16 Dec 2009 19:30:13 +0000 | |
changeset 190 | 16efb5711c6f |
parent 189 | a3631a999462 |
child 191 | 8c2c330e87f2 |
permissions | -rw-r--r-- |
94 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
3 |
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
|
4 |
||
141 | 5 |
\section{$n$-categories} |
94 | 6 |
\label{sec:ncats} |
7 |
||
108 | 8 |
\subsection{Definition of $n$-categories} |
9 |
||
94 | 10 |
Before proceeding, we need more appropriate definitions of $n$-categories, |
11 |
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
|
187 | 12 |
(As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of |
13 |
a `weak' $n$-category with `strong duality'.) |
|
94 | 14 |
|
141 | 15 |
The definitions presented below tie the categories more closely to the topology |
16 |
and avoid combinatorial questions about, for example, the minimal sufficient |
|
17 |
collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
|
18 |
For examples of topological origin, it is typically easy to show that they |
|
19 |
satisfy our axioms. |
|
20 |
For examples of a more purely algebraic origin, one would typically need the combinatorial |
|
21 |
results that we have avoided here. |
|
22 |
||
23 |
\medskip |
|
24 |
||
94 | 25 |
Consider first ordinary $n$-categories. |
26 |
We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
|
27 |
We must decide on the ``shape" of the $k$-morphisms. |
|
28 |
Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
|
29 |
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
|
30 |
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
|
31 |
and so on. |
|
32 |
(This allows for strict associativity.) |
|
145 | 33 |
Still other definitions \nn{need refs for all these; maybe the Leinster book \cite{MR2094071}} |
94 | 34 |
model the $k$-morphisms on more complicated combinatorial polyhedra. |
35 |
||
108 | 36 |
We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
37 |
the standard $k$-ball. |
|
94 | 38 |
In other words, |
39 |
||
187 | 40 |
\begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
41 |
For any $k$-manifold $X$ homeomorphic |
|
103 | 42 |
to the standard $k$-ball, we have a set of $k$-morphisms |
187 | 43 |
$\cC_k(X)$. |
44 |
\end{preliminary-axiom} |
|
94 | 45 |
|
103 | 46 |
Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
47 |
standard $k$-ball. |
|
48 |
We {\it do not} assume that it is equipped with a |
|
49 |
preferred homeomorphism to the standard $k$-ball. |
|
50 |
The same goes for ``a $k$-sphere" below. |
|
51 |
||
109 | 52 |
|
53 |
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
|
54 |
the boundary), we want a corresponding |
|
94 | 55 |
bijection of sets $f:\cC(X)\to \cC(Y)$. |
109 | 56 |
(This will imply ``strong duality", among other things.) |
94 | 57 |
So we replace the above with |
58 |
||
187 | 59 |
\begin{axiom}[Morphisms] |
60 |
\label{axiom:morphisms} |
|
61 |
For each $0 \le k \le n$, we have a functor $\cC_k$ from |
|
103 | 62 |
the category of $k$-balls and |
187 | 63 |
homeomorphisms to the category of sets and bijections. |
64 |
\end{axiom} |
|
65 |
||
94 | 66 |
|
67 |
(Note: We usually omit the subscript $k$.) |
|
68 |
||
187 | 69 |
We are so far being deliberately vague about what flavor of manifolds we are considering. |
94 | 70 |
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
71 |
They could be topological or PL or smooth. |
|
109 | 72 |
\nn{need to check whether this makes much difference --- see pseudo-isotopy below} |
94 | 73 |
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
74 |
to be fussier about corners.) |
|
75 |
For each flavor of manifold there is a corresponding flavor of $n$-category. |
|
76 |
We will concentrate of the case of PL unoriented manifolds. |
|
77 |
||
78 |
Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
|
79 |
of morphisms). |
|
80 |
The 0-sphere is unusual among spheres in that it is disconnected. |
|
81 |
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
|
109 | 82 |
(Actually, this is only true in the oriented case, with 1-morphsims parameterized |
83 |
by oriented 1-balls.) |
|
94 | 84 |
For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
85 |
\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
|
86 |
We prefer to combine the domain and range into a single entity which we call the |
|
87 |
boundary of a morphism. |
|
88 |
Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
|
89 |
||
187 | 90 |
\begin{axiom}[Boundaries (spheres)] |
91 |
For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
|
103 | 92 |
the category of $k$-spheres and |
187 | 93 |
homeomorphisms to the category of sets and bijections. |
94 |
\end{axiom} |
|
94 | 95 |
|
96 |
(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
|
97 |
||
187 | 98 |
\begin{axiom}[Boundaries (maps)] |
99 |
For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
|
100 |
These maps, for various $X$, comprise a natural transformation of functors. |
|
101 |
\end{axiom} |
|
94 | 102 |
|
103 |
(Note that the first ``$\bd$" above is part of the data for the category, |
|
104 |
while the second is the ordinary boundary of manifolds.) |
|
105 |
||
109 | 106 |
Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
94 | 107 |
|
108 |
Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
|
103 | 109 |
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
94 | 110 |
all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
111 |
(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
|
112 |
and all the structure maps of the $n$-category should be compatible with the auxiliary |
|
113 |
category structure. |
|
114 |
Note that this auxiliary structure is only in dimension $n$; |
|
115 |
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
|
116 |
||
117 |
\medskip |
|
109 | 118 |
\nn{ |
119 |
%At the moment I'm a little confused about orientations, and more specifically |
|
120 |
%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
|
121 |
Maybe need a discussion about what the boundary of a manifold with a |
|
122 |
structure (e.g. orientation) means. |
|
94 | 123 |
Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
124 |
Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
|
125 |
first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
|
126 |
equipped with an orientation of its once-stabilized tangent bundle. |
|
127 |
Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
|
128 |
their $k$ times stabilized tangent bundles. |
|
141 | 129 |
(cf. [Stolz and Teichner].) |
115 | 130 |
Probably should also have a framing of the stabilized dimensions in order to indicate which |
131 |
side the bounded manifold is on. |
|
94 | 132 |
For the moment just stick with unoriented manifolds.} |
133 |
\medskip |
|
134 |
||
135 |
We have just argued that the boundary of a morphism has no preferred splitting into |
|
136 |
domain and range, but the converse meets with our approval. |
|
137 |
That is, given compatible domain and range, we should be able to combine them into |
|
138 |
the full boundary of a morphism: |
|
139 |
||
187 | 140 |
\begin{axiom}[Boundary from domain and range] |
141 |
Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$, |
|
179 | 142 |
$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
94 | 143 |
Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
144 |
two maps $\bd: \cC(B_i)\to \cC(E)$. |
|
187 | 145 |
Then we have an injective map |
94 | 146 |
\[ |
147 |
\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
|
148 |
\] |
|
187 | 149 |
which is natural with respect to the actions of homeomorphisms. |
150 |
\end{axiom} |
|
94 | 151 |
|
179 | 152 |
\begin{figure}[!ht] |
186 | 153 |
$$ |
154 |
\begin{tikzpicture}[every label/.style={green}] |
|
155 |
\node[fill=black, circle, label=below:$E$](S) at (0,0) {}; |
|
156 |
\node[fill=black, circle, label=above:$E$](N) at (0,2) {}; |
|
157 |
\draw (S) arc (-90:90:1); |
|
158 |
\draw (N) arc (90:270:1); |
|
159 |
\node[left] at (-1,1) {$B_1$}; |
|
160 |
\node[right] at (1,1) {$B_2$}; |
|
161 |
\end{tikzpicture} |
|
162 |
$$ |
|
179 | 163 |
$$\mathfig{.4}{tempkw/blah3}$$ |
164 |
\caption{Combining two balls to get a full boundary}\label{blah3}\end{figure} |
|
165 |
||
94 | 166 |
Note that we insist on injectivity above. |
109 | 167 |
|
94 | 168 |
Let $\cC(S)_E$ denote the image of $\gl_E$. |
109 | 169 |
We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
170 |
||
110 | 171 |
We will call the projection $\cC(S)_E \to \cC(B_i)$ |
172 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
|
173 |
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
|
174 |
These restriction maps can be thought of as |
|
94 | 175 |
domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
176 |
||
103 | 177 |
If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
94 | 178 |
as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
179 |
||
180 |
Next we consider composition of morphisms. |
|
181 |
For $n$-categories which lack strong duality, one usually considers |
|
182 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
183 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
184 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
185 |
one general type of composition which can be in any ``direction". |
|
186 |
||
187 | 187 |
\begin{axiom}[Composition] |
188 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
|
179 | 189 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 190 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 191 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
192 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
193 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
194 |
Then (axiom) we have a map |
|
195 |
\[ |
|
196 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
197 |
\] |
|
198 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
199 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
200 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
187 | 201 |
(For $k=n$, see below.) |
202 |
\end{axiom} |
|
94 | 203 |
|
179 | 204 |
\begin{figure}[!ht] |
205 |
$$\mathfig{.4}{tempkw/blah5}$$ |
|
206 |
\caption{From two balls to one ball}\label{blah5}\end{figure} |
|
207 |
||
187 | 208 |
\begin{axiom}[Strict associativity] |
209 |
The composition (gluing) maps above are strictly associative. |
|
210 |
\end{axiom} |
|
102 | 211 |
|
179 | 212 |
\begin{figure}[!ht] |
213 |
$$\mathfig{.65}{tempkw/blah6}$$ |
|
214 |
\caption{An example of strict associativity}\label{blah6}\end{figure} |
|
215 |
||
109 | 216 |
Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
110 | 217 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
218 |
a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
|
219 |
Compositions of boundary and restriction maps will also be called restriction maps. |
|
220 |
For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
221 |
restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
222 |
||
223 |
%More notation and terminology: |
|
224 |
%We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction} |
|
225 |
%map |
|
109 | 226 |
|
102 | 227 |
The above two axioms are equivalent to the following axiom, |
228 |
which we state in slightly vague form. |
|
229 |
||
230 |
\xxpar{Multi-composition:} |
|
231 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
232 |
into small $k$-balls, there is a |
|
233 |
map from an appropriate subset (like a fibered product) |
|
234 |
of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
|
95 | 235 |
and these various $m$-fold composition maps satisfy an |
179 | 236 |
operad-type strict associativity condition (Figure \ref{blah7}).} |
237 |
||
238 |
\begin{figure}[!ht] |
|
239 |
$$\mathfig{.8}{tempkw/blah7}$$ |
|
240 |
\caption{Operadish composition and associativity}\label{blah7}\end{figure} |
|
95 | 241 |
|
242 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
243 |
||
187 | 244 |
\begin{axiom}[Product (identity) morphisms] |
245 |
Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |
|
95 | 246 |
Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
247 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
|
248 |
\[ \xymatrix{ |
|
96 | 249 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
95 | 250 |
X \ar[r]^{f} & X' |
251 |
} \] |
|
109 | 252 |
commutes, then we have |
253 |
\[ |
|
254 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
255 |
\] |
|
256 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
257 |
\[ |
|
258 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
259 |
\] |
|
260 |
and |
|
261 |
\[ |
|
262 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
263 |
\] |
|
122 | 264 |
\nn{if pinched boundary, then remove first case above} |
109 | 265 |
Product morphisms are associative: |
266 |
\[ |
|
267 |
(a\times D)\times D' = a\times (D\times D') . |
|
268 |
\] |
|
269 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
270 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
110 | 271 |
Product morphisms are compatible with restriction: |
272 |
\[ |
|
273 |
\res_{X\times E}(a\times D) = a\times E |
|
274 |
\] |
|
275 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
187 | 276 |
\end{axiom} |
95 | 277 |
|
110 | 278 |
\nn{need even more subaxioms for product morphisms?} |
95 | 279 |
|
122 | 280 |
\nn{Almost certainly we need a little more than the above axiom. |
281 |
More specifically, in order to bootstrap our way from the top dimension |
|
282 |
properties of identity morphisms to low dimensions, we need regular products, |
|
283 |
pinched products and even half-pinched products. |
|
142 | 284 |
I'm not sure what the best way to cleanly axiomatize the properties of these various |
285 |
products is. |
|
122 | 286 |
For the moment, I'll assume that all flavors of the product are at |
287 |
our disposal, and I'll plan on revising the axioms later.} |
|
288 |
||
128 | 289 |
\nn{current idea for fixing this: make the above axiom a ``preliminary version" |
290 |
(as we have already done with some of the other axioms), then state the official |
|
291 |
axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
|
292 |
one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
|
293 |
the alternative is to give some sort of local criterion for what's allowed. |
|
294 |
state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
|
295 |
} |
|
296 |
||
95 | 297 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
298 |
The last axiom (below), concerning actions of |
|
299 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
300 |
||
301 |
We start with the plain $n$-category case. |
|
302 |
||
187 | 303 |
\begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$} |
304 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
95 | 305 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
187 | 306 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
307 |
\end{preliminary-axiom} |
|
96 | 308 |
|
174 | 309 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 310 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 311 |
Let $J$ be a 1-ball (interval). |
312 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 313 |
(Here we use the ``pinched" version of $Y\times J$. |
314 |
\nn{need notation for this}) |
|
96 | 315 |
We define a map |
316 |
\begin{eqnarray*} |
|
317 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
318 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
319 |
\end{eqnarray*} |
|
142 | 320 |
(See Figure \ref{glue-collar}.) |
189 | 321 |
\begin{figure}[!ht] |
322 |
\begin{equation*} |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
323 |
\begin{tikzpicture} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
324 |
\def\rad{1} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
325 |
\def\srad{0.75} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
326 |
\def\gap{4.5} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
327 |
\foreach \i in {0, 1, 2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
328 |
\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
|
329 |
\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
|
330 |
\foreach \n in {1,2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
331 |
\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
|
332 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
333 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
334 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
335 |
\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
336 |
\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
|
337 |
\end{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
338 |
\node[right=1mm] at (0.east) {$a$}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
339 |
\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
|
340 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
341 |
\draw (1-small) circle (\srad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
342 |
\foreach \theta in {90, 72, ..., -90} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
343 |
\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
344 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
345 |
\filldraw[fill=white] (1) circle (\rad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
346 |
\foreach \n in {1,2} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
347 |
\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
|
348 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
349 |
\node[below] at (1-small.south) {$a \times J$}; |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
350 |
\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
|
351 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
352 |
\begin{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
353 |
\path[clip] (2) circle (\rad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
354 |
\draw[clip] (2.east) circle (\srad); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
355 |
\foreach \y in {1, 0.86, ..., -1} { |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
356 |
\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
357 |
} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
358 |
\end{scope} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
359 |
\end{tikzpicture} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
360 |
\end{equation*} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
361 |
\begin{equation*} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
362 |
\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 | 363 |
\end{equation*} |
364 |
||
365 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
174 | 366 |
We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
367 |
\nn{bad terminology; fix it later} |
|
368 |
\nn{also need to make clear that plain old isotopic to the identity implies |
|
369 |
extended isotopic} |
|
97 | 370 |
\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
371 |
extended isotopies are also plain isotopies, so |
|
372 |
no extension necessary} |
|
96 | 373 |
It can be thought of as the action of the inverse of |
374 |
a map which projects a collar neighborhood of $Y$ onto $Y$. |
|
375 |
||
376 |
The revised axiom is |
|
377 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
378 |
\stepcounter{axiom} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
379 |
\begin{axiom-numbered}{\arabic{axiom}a}{Extended isotopy invariance in dimension $n$} |
187 | 380 |
\label{axiom:extended-isotopies} |
381 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
174 | 382 |
to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
187 | 383 |
Then $f$ acts trivially on $\cC(X)$. |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
384 |
\end{axiom-numbered} |
96 | 385 |
|
386 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
94 | 387 |
|
97 | 388 |
\smallskip |
389 |
||
390 |
For $A_\infty$ $n$-categories, we replace |
|
391 |
isotopy invariance with the requirement that families of homeomorphisms act. |
|
392 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
|
393 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
394 |
\begin{axiom-numbered}{\arabic{axiom}b}{Families of homeomorphisms act in dimension $n$} |
187 | 395 |
For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
97 | 396 |
\[ |
397 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
398 |
\] |
|
399 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
|
400 |
which fix $\bd X$. |
|
401 |
These action maps are required to be associative up to homotopy |
|
402 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
403 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
404 |
\nn{repeat diagram here?} |
|
187 | 405 |
\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
|
406 |
\end{axiom-numbered} |
97 | 407 |
|
408 |
We should strengthen the above axiom to apply to families of extended homeomorphisms. |
|
109 | 409 |
To do this we need to explain how extended homeomorphisms form a topological space. |
97 | 410 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
411 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
|
412 |
\nn{need to also say something about collaring homeomorphisms.} |
|
413 |
\nn{this paragraph needs work.} |
|
414 |
||
103 | 415 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
416 |
into a plain $n$-category (enriched over graded groups). |
|
97 | 417 |
\nn{say more here?} |
418 |
In the other direction, if we enrich over topological spaces instead of chain complexes, |
|
419 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
|
420 |
instead of $C_*(\Homeo_\bd(X))$. |
|
421 |
Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
|
422 |
type $A_\infty$ $n$-category. |
|
423 |
||
99 | 424 |
\medskip |
97 | 425 |
|
99 | 426 |
The alert reader will have already noticed that our definition of (plain) $n$-category |
427 |
is extremely similar to our definition of topological fields. |
|
142 | 428 |
The main difference is that for the $n$-category definition we restrict our attention to balls |
99 | 429 |
(and their boundaries), while for fields we consider all manifolds. |
142 | 430 |
(A minor difference is that in the category definition we directly impose isotopy |
431 |
invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields |
|
432 |
but then mod out by local relations which imply isotopy invariance.) |
|
99 | 433 |
Thus a system of fields determines an $n$-category simply by restricting our attention to |
434 |
balls. |
|
142 | 435 |
This $n$-category can be thought of as the local part of the fields. |
99 | 436 |
Conversely, given an $n$-category we can construct a system of fields via |
142 | 437 |
a colimit construction; see below. |
99 | 438 |
|
142 | 439 |
%\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
440 |
%of fields. |
|
441 |
%The universal (colimit) construction becomes our generalized definition of blob homology. |
|
442 |
%Need to explain how it relates to the old definition.} |
|
97 | 443 |
|
95 | 444 |
\medskip |
445 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
446 |
\subsection{Examples of $n$-categories} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
447 |
|
101 | 448 |
\nn{these examples need to be fleshed out a bit more} |
449 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
450 |
We know describe several classes of examples of $n$-categories satisfying our axioms. |
101 | 451 |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
452 |
\begin{example}{Maps to a space} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
453 |
\label{ex:maps-to-a-space}% |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
454 |
Fix $F$ a closed $m$-manifold (keep in mind the case where $F$ is a point). Fix a `target space' $T$, any topological space. |
101 | 455 |
For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
456 |
all maps from $X\times F$ to $T$. |
|
457 |
For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
|
103 | 458 |
homotopies fixed on $\bd X \times F$. |
101 | 459 |
(Note that homotopy invariance implies isotopy invariance.) |
460 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
461 |
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
|
462 |
\end{example} |
101 | 463 |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
464 |
\begin{example}{Linearized, twisted, maps to a space} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
465 |
\label{ex:linearized-maps-to-a-space}% |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
466 |
We can linearize the above example as follows. |
101 | 467 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
468 |
(e.g.\ the trivial cocycle). |
|
469 |
For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
|
470 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
|
471 |
the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
|
472 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
|
473 |
$h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
|
474 |
\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
|
475 |
\end{example} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
476 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
477 |
\begin{itemize} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
478 |
|
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
479 |
\item \nn{Continue converting these into examples} |
101 | 480 |
|
481 |
\item Given a traditional $n$-category $C$ (with strong duality etc.), |
|
482 |
define $\cC(X)$ (with $\dim(X) < n$) |
|
483 |
to be the set of all $C$-labeled sub cell complexes of $X$. |
|
142 | 484 |
(See Subsection \ref{sec:fields}.) |
101 | 485 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
486 |
combinations of $C$-labeled sub cell complexes of $X$ |
|
487 |
modulo the kernel of the evaluation map. |
|
488 |
Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
|
489 |
and with the same labeling as $a$. |
|
102 | 490 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
491 |
Define $\cC(X)$, for $\dim(X) < n$, |
|
492 |
to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
|
493 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
|
494 |
to be the dual Hilbert space $A(X\times F; c)$. |
|
101 | 495 |
\nn{refer elsewhere for details?} |
496 |
||
497 |
\item Variation on the above examples: |
|
103 | 498 |
We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
101 | 499 |
for example product boundary conditions or take the union over all boundary conditions. |
142 | 500 |
%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
501 |
%to think of these guys as affording a representation |
|
502 |
%of the $n{+}1$-category associated to $\bd F$.} |
|
101 | 503 |
|
142 | 504 |
\item Here's our version of the bordism $n$-category. |
505 |
For a $k$-ball $X$, $k<n$, define $\cC(X)$ to be the set of all $k$-dimensional |
|
506 |
submanifolds $W$ of $X\times \r^\infty$ such that the projection $W \to X$ is transverse |
|
507 |
to $\bd X$. |
|
508 |
For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
|
509 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
|
510 |
$W\to W'$ which restricts to the identity on the boundary. |
|
511 |
||
143 | 512 |
\item \nn{sphere modules; ref to below} |
125 | 513 |
|
101 | 514 |
\end{itemize} |
515 |
||
516 |
||
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
517 |
We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 518 |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
519 |
\begin{example}{Chains of maps to a space} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
520 |
We can modify Example \ref{ex:maps-to-a-space} above by defining $\cC(X; c)$ for an $n$-ball $X$ to be the chain complex |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
521 |
$C_*(\Maps_c(X\times F \to T))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 522 |
and $C_*$ denotes singular chains. |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
523 |
\end{example} |
101 | 524 |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
525 |
\begin{example}{Blob complexes of balls (with a fiber)} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
526 |
Fix an $m$-dimensional manifold $F$. |
101 | 527 |
Given a plain $n$-category $C$, |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
528 |
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
|
529 |
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
|
530 |
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
|
531 |
\end{example} |
101 | 532 |
|
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
533 |
\begin{defn} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
534 |
\nn{should add $\infty$ version of bordism $n$-cat} |
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
535 |
\end{defn} |
95 | 536 |
|
108 | 537 |
|
538 |
||
539 |
||
540 |
||
541 |
||
542 |
\subsection{From $n$-categories to systems of fields} |
|
113 | 543 |
\label{ss:ncat_fields} |
108 | 544 |
|
545 |
We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
|
546 |
||
547 |
Let $W$ be a $k$-manifold, $1\le k \le n$. |
|
548 |
We will define a set $\cC(W)$. |
|
549 |
(If $k = n$ and our $k$-categories are enriched, then |
|
550 |
$\cC(W)$ will have additional structure; see below.) |
|
551 |
$\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
|
552 |
which we define next. |
|
553 |
||
142 | 554 |
Define a permissible decomposition of $W$ to be a cell decomposition |
108 | 555 |
\[ |
556 |
W = \bigcup_a X_a , |
|
557 |
\] |
|
142 | 558 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
108 | 559 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
560 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
561 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
562 |
(The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
|
119 | 563 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
564 |
See Figure \ref{partofJfig}.) |
|
565 |
||
566 |
\begin{figure}[!ht] |
|
567 |
\begin{equation*} |
|
568 |
\mathfig{.63}{tempkw/zz2} |
|
569 |
\end{equation*} |
|
570 |
\caption{A small part of $\cJ(W)$} |
|
571 |
\label{partofJfig} |
|
572 |
\end{figure} |
|
573 |
||
108 | 574 |
|
575 |
$\cC$ determines |
|
576 |
a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
|
577 |
(possibly with additional structure if $k=n$). |
|
578 |
For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
|
579 |
\[ |
|
580 |
\psi_\cC(x) \sub \prod_a \cC(X_a) |
|
581 |
\] |
|
582 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
583 |
$X_a$ all agree. |
|
584 |
(Think fibered product.) |
|
585 |
If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
|
586 |
via the composition maps of $\cC$. |
|
112 | 587 |
(If $\dim(W) = n$ then we need to also make use of the monoidal |
588 |
product in the enriching category. |
|
589 |
\nn{should probably be more explicit here}) |
|
108 | 590 |
|
591 |
Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
|
142 | 592 |
When $k<n$ or $k=n$ and we are in the plain (non-$A_\infty$) case, this means that |
112 | 593 |
for each decomposition $x$ there is a map |
108 | 594 |
$\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
595 |
above, and $\cC(W)$ is universal with respect to these properties. |
|
142 | 596 |
When $k=n$ and we are in the $A_\infty$ case, it means |
597 |
homotopy colimit. |
|
112 | 598 |
|
599 |
More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
|
600 |
\[ |
|
601 |
\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
|
602 |
\] |
|
603 |
where $K$ is generated by all things of the form $a - g(a)$, where |
|
604 |
$a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
|
605 |
\to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
|
111 | 606 |
|
112 | 607 |
In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
608 |
is as follows. |
|
142 | 609 |
%\nn{should probably rewrite this to be compatible with some standard reference} |
113 | 610 |
Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
112 | 611 |
Such sequences (for all $m$) form a simplicial set. |
612 |
Let |
|
613 |
\[ |
|
614 |
V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
|
615 |
\] |
|
113 | 616 |
where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$. |
112 | 617 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
618 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
|
619 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
620 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
|
621 |
\[ |
|
622 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) , |
|
623 |
\] |
|
624 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
|
625 |
is the usual map. |
|
626 |
\nn{need to say this better} |
|
627 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
628 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
629 |
of $A_\infty$ category} |
|
108 | 630 |
|
113 | 631 |
We will call $m$ the filtration degree of the complex. |
632 |
We can think of this construction as starting with a disjoint copy of a complex for each |
|
633 |
permissible decomposition (filtration degree 0). |
|
634 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
635 |
(filtration degree 1). |
|
636 |
Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2). |
|
637 |
And so on. |
|
638 |
||
108 | 639 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
640 |
||
641 |
It is easy to see that |
|
642 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
|
643 |
comprise a natural transformation of functors. |
|
644 |
||
645 |
\nn{need to finish explaining why we have a system of fields; |
|
646 |
need to say more about ``homological" fields? |
|
647 |
(actions of homeomorphisms); |
|
648 |
define $k$-cat $\cC(\cdot\times W)$} |
|
649 |
||
650 |
||
651 |
||
652 |
\subsection{Modules} |
|
95 | 653 |
|
101 | 654 |
Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
655 |
a.k.a.\ actions). |
|
102 | 656 |
The definition will be very similar to that of $n$-categories. |
109 | 657 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
110 | 658 |
\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
102 | 659 |
|
104 | 660 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 661 |
in the context of an $m{+}1$-dimensional TQFT. |
662 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
663 |
This will be explained in more detail as we present the axioms. |
|
664 |
||
665 |
Fix an $n$-category $\cC$. |
|
666 |
||
667 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
668 |
(standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
|
669 |
We call $B$ the ball and $N$ the marking. |
|
670 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
671 |
restricts to a homeomorphism of markings. |
|
672 |
||
673 |
\xxpar{Module morphisms} |
|
674 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
|
675 |
the category of marked $k$-balls and |
|
676 |
homeomorphisms to the category of sets and bijections.} |
|
677 |
||
678 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
679 |
||
104 | 680 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
681 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
682 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
683 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
684 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
685 |
(The union is along $N\times \bd W$.) |
|
110 | 686 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
687 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 688 |
|
182 | 689 |
\begin{figure}[!ht] |
690 |
$$\mathfig{.8}{tempkw/blah15}$$ |
|
691 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
|
692 |
||
103 | 693 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
694 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 695 |
|
696 |
\xxpar{Module boundaries, part 1:} |
|
697 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
|
104 | 698 |
the category of marked $k$-hemispheres and |
102 | 699 |
homeomorphisms to the category of sets and bijections.} |
700 |
||
104 | 701 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
702 |
||
102 | 703 |
\xxpar{Module boundaries, part 2:} |
704 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
|
705 |
These maps, for various $M$, comprise a natural transformation of functors.} |
|
706 |
||
110 | 707 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 708 |
|
709 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
710 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
711 |
and $c\in \cC(\bd M)$. |
|
712 |
||
713 |
\xxpar{Module domain $+$ range $\to$ boundary:} |
|
714 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
|
104 | 715 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
716 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
|
717 |
two maps $\bd: \cM(M_i)\to \cM(E)$. |
|
102 | 718 |
Then (axiom) we have an injective map |
719 |
\[ |
|
720 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
|
721 |
\] |
|
722 |
which is natural with respect to the actions of homeomorphisms.} |
|
723 |
||
110 | 724 |
Let $\cM(H)_E$ denote the image of $\gl_E$. |
725 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
|
726 |
||
727 |
||
103 | 728 |
\xxpar{Axiom yet to be named:} |
729 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
|
730 |
$\cM(H)\to \cC(H)$. |
|
731 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
|
732 |
These maps comprise a natural transformation of functors.} |
|
102 | 733 |
|
103 | 734 |
Note that combining the various boundary and restriction maps above |
110 | 735 |
(for both modules and $n$-categories) |
103 | 736 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
737 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 738 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
739 |
cutting submanifolds). |
|
103 | 740 |
This fact will be used below. |
102 | 741 |
|
104 | 742 |
In our example, the various restriction and gluing maps above come from |
743 |
restricting and gluing maps into $T$. |
|
744 |
||
745 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 746 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 747 |
(See Figure \ref{zzz3}.) |
103 | 748 |
|
119 | 749 |
\begin{figure}[!ht] |
750 |
\begin{equation*} |
|
751 |
\mathfig{.63}{tempkw/zz3} |
|
752 |
\end{equation*} |
|
753 |
\caption{Module composition (top); $n$-category action (bottom)} |
|
754 |
\label{zzz3} |
|
755 |
\end{figure} |
|
756 |
||
757 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 758 |
|
759 |
\xxpar{Module composition:} |
|
760 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
|
761 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
|
762 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
763 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
764 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
765 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
766 |
Then (axiom) we have a map |
|
767 |
\[ |
|
768 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
769 |
\] |
|
770 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
771 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
772 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
773 |
(For $k=n$, see below.)} |
|
774 |
||
119 | 775 |
|
776 |
||
103 | 777 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
778 |
module morphism. |
|
779 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
780 |
||
781 |
\xxpar{$n$-category action:} |
|
782 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
|
783 |
$X$ is a plain $k$-ball, |
|
784 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
785 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
786 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
787 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
788 |
Then (axiom) we have a map |
|
789 |
\[ |
|
790 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
791 |
\] |
|
792 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
793 |
to the intersection of the boundaries of $X$ and $M'$. |
|
794 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
795 |
(For $k=n$, see below.)} |
|
796 |
||
797 |
\xxpar{Module strict associativity:} |
|
798 |
{The composition and action maps above are strictly associative.} |
|
799 |
||
110 | 800 |
Note that the above associativity axiom applies to mixtures of module composition, |
801 |
action maps and $n$-category composition. |
|
119 | 802 |
See Figure \ref{zzz1b}. |
803 |
||
804 |
\begin{figure}[!ht] |
|
805 |
\begin{equation*} |
|
806 |
\mathfig{1}{tempkw/zz1b} |
|
807 |
\end{equation*} |
|
808 |
\caption{Two examples of mixed associativity} |
|
809 |
\label{zzz1b} |
|
810 |
\end{figure} |
|
811 |
||
110 | 812 |
|
813 |
The above three axioms are equivalent to the following axiom, |
|
103 | 814 |
which we state in slightly vague form. |
815 |
\nn{need figure for this} |
|
816 |
||
817 |
\xxpar{Module multi-composition:} |
|
818 |
{Given any decomposition |
|
819 |
\[ |
|
820 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
821 |
\] |
|
822 |
of a marked $k$-ball $M$ |
|
823 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
824 |
map from an appropriate subset (like a fibered product) |
|
825 |
of |
|
826 |
\[ |
|
827 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
828 |
\] |
|
829 |
to $\cM(M)$, |
|
830 |
and these various multifold composition maps satisfy an |
|
831 |
operad-type strict associativity condition.} |
|
832 |
||
833 |
(The above operad-like structure is analogous to the swiss cheese operad |
|
146 | 834 |
\cite{MR1718089}.) |
103 | 835 |
\nn{need to double-check that this is true.} |
836 |
||
837 |
\xxpar{Module product (identity) morphisms:} |
|
838 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
|
839 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
|
840 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
|
841 |
\[ \xymatrix{ |
|
842 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
|
843 |
M \ar[r]^{f} & M' |
|
844 |
} \] |
|
845 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
846 |
||
111 | 847 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
103 | 848 |
|
110 | 849 |
\nn{** marker --- resume revising here **} |
850 |
||
103 | 851 |
There are two alternatives for the next axiom, according whether we are defining |
852 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
853 |
In the plain case we require |
|
854 |
||
185 | 855 |
\xxpar{Extended isotopy invariance in dimension $n$:} |
103 | 856 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
175 | 857 |
to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
103 | 858 |
Then $f$ acts trivially on $\cM(M)$.} |
859 |
||
860 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
861 |
||
862 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
863 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
864 |
on $\bd B \setmin N$. |
|
865 |
||
866 |
For $A_\infty$ modules we require |
|
867 |
||
868 |
\xxpar{Families of homeomorphisms act.} |
|
869 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
|
870 |
\[ |
|
871 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
872 |
\] |
|
873 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
874 |
which fix $\bd M$. |
|
875 |
These action maps are required to be associative up to homotopy |
|
876 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
877 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
878 |
\nn{repeat diagram here?} |
|
879 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
880 |
||
881 |
\medskip |
|
102 | 882 |
|
104 | 883 |
Note that the above axioms imply that an $n$-category module has the structure |
884 |
of an $n{-}1$-category. |
|
885 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
886 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
887 |
above the non-marked boundary component of $J$. |
|
888 |
\nn{give figure for this, or say more?} |
|
889 |
Then $\cE$ has the structure of an $n{-}1$-category. |
|
102 | 890 |
|
105 | 891 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
892 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
893 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
894 |
of marked 1-balls, call them left-marked and right-marked, |
|
895 |
and hence there are two types of modules, call them right modules and left modules. |
|
896 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
897 |
there is no left/right module distinction. |
|
898 |
||
130 | 899 |
\medskip |
900 |
||
901 |
Examples of modules: |
|
902 |
\begin{itemize} |
|
142 | 903 |
\item \nn{examples from TQFTs} |
904 |
\item \nn{for maps to $T$, can restrict to subspaces of $T$;} |
|
130 | 905 |
\end{itemize} |
108 | 906 |
|
907 |
\subsection{Modules as boundary labels} |
|
112 | 908 |
\label{moddecss} |
108 | 909 |
|
910 |
Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
143 | 911 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
912 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
913 |
||
914 |
%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
915 |
%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
916 |
%component $\bd_i W$ of $W$. |
|
917 |
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
108 | 918 |
|
919 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
|
920 |
\nn{give ref} |
|
921 |
(If $k = n$ and our $k$-categories are enriched, then |
|
922 |
$\cC(W, \cN)$ will have additional structure; see below.) |
|
923 |
||
924 |
Define a permissible decomposition of $W$ to be a decomposition |
|
925 |
\[ |
|
926 |
W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
|
927 |
\] |
|
928 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
929 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
143 | 930 |
with $M_{ib}\cap Y_i$ being the marking. |
931 |
(See Figure \ref{mblabel}.) |
|
932 |
\begin{figure}[!ht]\begin{equation*} |
|
933 |
\mathfig{.9}{tempkw/mblabel} |
|
934 |
\end{equation*}\caption{A permissible decomposition of a manifold |
|
146 | 935 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
108 | 936 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
937 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
938 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
939 |
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
|
940 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
|
941 |
||
942 |
$\cN$ determines |
|
943 |
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
|
944 |
(possibly with additional structure if $k=n$). |
|
945 |
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
|
946 |
\[ |
|
111 | 947 |
\psi_\cN(x) \sub (\prod_a \cC(X_a)) \times (\prod_{ib} \cN_i(M_{ib})) |
108 | 948 |
\] |
949 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
950 |
$X_a$ and $M_{ib}$ all agree. |
|
951 |
(Think fibered product.) |
|
952 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
|
953 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
954 |
||
955 |
Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
|
143 | 956 |
(Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
957 |
homotopy colimit.) |
|
108 | 958 |
|
143 | 959 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
960 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
961 |
$D\times Y_i \sub \bd(D\times W)$. |
|
112 | 962 |
|
143 | 963 |
It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
964 |
has the structure of an $n{-}k$-category. |
|
144 | 965 |
|
966 |
\medskip |
|
967 |
||
968 |
||
969 |
%\subsection{Tensor products} |
|
108 | 970 |
|
144 | 971 |
We will use a simple special case of the above |
972 |
construction to define tensor products |
|
973 |
of modules. |
|
974 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
975 |
(If $k=1$ and manifolds are oriented, then one should be |
|
976 |
a left module and the other a right module.) |
|
977 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
978 |
Define the tensor product of $\cM_1$ and $\cM_2$ to be the |
|
979 |
$n{-}1$-category $\cT(J, \cM_1, \cM_2)$, |
|
980 |
\[ |
|
981 |
\cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) . |
|
982 |
\] |
|
983 |
This of course depends (functorially) |
|
984 |
on the choice of 1-ball $J$. |
|
105 | 985 |
|
144 | 986 |
We will define a more general self tensor product (categorified coend) below. |
987 |
||
112 | 988 |
|
144 | 989 |
|
990 |
||
991 |
%\nn{what about self tensor products /coends ?} |
|
105 | 992 |
|
108 | 993 |
\nn{maybe ``tensor product" is not the best name?} |
994 |
||
144 | 995 |
%\nn{start with (less general) tensor products; maybe change this later} |
106 | 996 |
|
107 | 997 |
|
998 |
||
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
999 |
\subsection{The $n{+}1$-category of sphere modules} |
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1000 |
|
155 | 1001 |
In this subsection we define an $n{+}1$-category of ``sphere modules" whose objects |
1002 |
correspond to $n$-categories. |
|
1003 |
This is a version of the familiar algebras-bimodules-intertwinors 2-category. |
|
1004 |
(Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule, |
|
1005 |
since a 0-sphere has an obvious bi-ness. |
|
1006 |
This is much less true for higher dimensional spheres, |
|
1007 |
so we prefer the term ``sphere module" for the general case.) |
|
144 | 1008 |
|
107 | 1009 |
|
1010 |
||
144 | 1011 |
\nn{need to assume a little extra structure to define the top ($n+1$) part (?)} |
101 | 1012 |
|
1013 |
\medskip |
|
1014 |
\hrule |
|
1015 |
\medskip |
|
1016 |
||
95 | 1017 |
\nn{to be continued...} |
101 | 1018 |
\medskip |
98 | 1019 |
|
1020 |
||
1021 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
|
1022 |
a separate paper): |
|
1023 |
\begin{itemize} |
|
1024 |
\item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
|
1025 |
\item conversely, our def implies other defs |
|
105 | 1026 |
\item do same for modules; maybe an appendix on relating topological |
1027 |
vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
|
98 | 1028 |
\item traditional $A_\infty$ 1-cat def implies our def |
99 | 1029 |
\item ... and vice-versa (already done in appendix) |
98 | 1030 |
\item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
1031 |
\item spell out what difference (if any) Top vs PL vs Smooth makes |
|
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1032 |
\item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
1033 |
a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
130 | 1034 |
\item morphisms of modules; show that it's adjoint to tensor product |
139 | 1035 |
(need to define dual module for this) |
1036 |
\item functors |
|
98 | 1037 |
\end{itemize} |
1038 |
||
134 | 1039 |
\nn{Some salvaged paragraphs that we might want to work back in:} |
1040 |
\hrule |
|
98 | 1041 |
|
134 | 1042 |
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.) |
1043 |
||
1044 |
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 |
|
1045 |
\begin{align*} |
|
1046 |
\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)), |
|
1047 |
\end{align*} |
|
1048 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
|
1049 |
||
1050 |
We now give two motivating examples, as theorems constructing other homological systems of fields, |
|
1051 |
||
1052 |
||
1053 |
\begin{thm} |
|
1054 |
For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as |
|
1055 |
\begin{equation*} |
|
1056 |
\Xi(M) = \CM{M}{X}. |
|
1057 |
\end{equation*} |
|
1058 |
\end{thm} |
|
1059 |
||
1060 |
\begin{thm} |
|
1061 |
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 |
|
1062 |
\begin{equation*} |
|
1063 |
\cF^{\times F}(M) = \cB_*(M \times F, \cF). |
|
1064 |
\end{equation*} |
|
1065 |
\end{thm} |
|
1066 |
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. |
|
1067 |
||
1068 |
||
1069 |
In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields. |
|
1070 |
||
1071 |
||
1072 |
\begin{thm} |
|
1073 |
\begin{equation*} |
|
1074 |
\cB_*(M, \Xi) \iso \Xi(M) |
|
1075 |
\end{equation*} |
|
1076 |
\end{thm} |
|
1077 |
||
1078 |
\begin{thm}[Product formula] |
|
1079 |
Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields, |
|
1080 |
there is a quasi-isomorphism |
|
1081 |
\begin{align*} |
|
1082 |
\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) |
|
1083 |
\end{align*} |
|
1084 |
\end{thm} |
|
1085 |
||
1086 |
\begin{question} |
|
1087 |
Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
|
1088 |
\end{question} |
|
1089 |
||
1090 |
\hrule |