author | scott@6e1638ff-ae45-0410-89bd-df963105f760 |
Mon, 26 Oct 2009 17:14:35 +0000 | |
changeset 135 | b15dafe85ee1 |
parent 134 | 395bd663e20d |
child 139 | 57291331fd82 |
permissions | -rw-r--r-- |
94 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
3 |
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
|
4 |
||
5 |
\section{$n$-categories (maybe)} |
|
6 |
\label{sec:ncats} |
|
7 |
||
8 |
\nn{experimental section. maybe this should be rolled into other sections. |
|
9 |
maybe it should be split off into a separate paper.} |
|
10 |
||
108 | 11 |
\subsection{Definition of $n$-categories} |
12 |
||
94 | 13 |
Before proceeding, we need more appropriate definitions of $n$-categories, |
14 |
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
|
15 |
(As is the case throughout this paper, by ``$n$-category" we mean |
|
16 |
a weak $n$-category with strong duality.) |
|
17 |
||
18 |
Consider first ordinary $n$-categories. |
|
19 |
We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
|
20 |
We must decide on the ``shape" of the $k$-morphisms. |
|
21 |
Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
|
22 |
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
|
23 |
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
|
24 |
and so on. |
|
25 |
(This allows for strict associativity.) |
|
26 |
Still other definitions \nn{need refs for all these; maybe the Leinster book} |
|
27 |
model the $k$-morphisms on more complicated combinatorial polyhedra. |
|
28 |
||
108 | 29 |
We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
30 |
the standard $k$-ball. |
|
94 | 31 |
In other words, |
32 |
||
102 | 33 |
\xxpar{Morphisms (preliminary version):} |
34 |
{For any $k$-manifold $X$ homeomorphic |
|
103 | 35 |
to the standard $k$-ball, we have a set of $k$-morphisms |
109 | 36 |
$\cC_k(X)$.} |
94 | 37 |
|
103 | 38 |
Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
39 |
standard $k$-ball. |
|
40 |
We {\it do not} assume that it is equipped with a |
|
41 |
preferred homeomorphism to the standard $k$-ball. |
|
42 |
The same goes for ``a $k$-sphere" below. |
|
43 |
||
109 | 44 |
|
45 |
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
|
46 |
the boundary), we want a corresponding |
|
94 | 47 |
bijection of sets $f:\cC(X)\to \cC(Y)$. |
109 | 48 |
(This will imply ``strong duality", among other things.) |
94 | 49 |
So we replace the above with |
50 |
||
102 | 51 |
\xxpar{Morphisms:} |
52 |
{For each $0 \le k \le n$, we have a functor $\cC_k$ from |
|
103 | 53 |
the category of $k$-balls and |
94 | 54 |
homeomorphisms to the category of sets and bijections.} |
55 |
||
56 |
(Note: We usually omit the subscript $k$.) |
|
57 |
||
58 |
We are being deliberately vague about what flavor of manifolds we are considering. |
|
59 |
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
|
60 |
They could be topological or PL or smooth. |
|
109 | 61 |
\nn{need to check whether this makes much difference --- see pseudo-isotopy below} |
94 | 62 |
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
63 |
to be fussier about corners.) |
|
64 |
For each flavor of manifold there is a corresponding flavor of $n$-category. |
|
65 |
We will concentrate of the case of PL unoriented manifolds. |
|
66 |
||
67 |
Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
|
68 |
of morphisms). |
|
69 |
The 0-sphere is unusual among spheres in that it is disconnected. |
|
70 |
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
|
109 | 71 |
(Actually, this is only true in the oriented case, with 1-morphsims parameterized |
72 |
by oriented 1-balls.) |
|
94 | 73 |
For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
74 |
\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
|
75 |
We prefer to combine the domain and range into a single entity which we call the |
|
76 |
boundary of a morphism. |
|
77 |
Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
|
78 |
||
79 |
\xxpar{Boundaries (domain and range), part 1:} |
|
80 |
{For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
|
103 | 81 |
the category of $k$-spheres and |
94 | 82 |
homeomorphisms to the category of sets and bijections.} |
83 |
||
84 |
(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
|
85 |
||
86 |
\xxpar{Boundaries, part 2:} |
|
103 | 87 |
{For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
94 | 88 |
These maps, for various $X$, comprise a natural transformation of functors.} |
89 |
||
90 |
(Note that the first ``$\bd$" above is part of the data for the category, |
|
91 |
while the second is the ordinary boundary of manifolds.) |
|
92 |
||
109 | 93 |
Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
94 | 94 |
|
95 |
Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
|
103 | 96 |
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
94 | 97 |
all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
98 |
(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
|
99 |
and all the structure maps of the $n$-category should be compatible with the auxiliary |
|
100 |
category structure. |
|
101 |
Note that this auxiliary structure is only in dimension $n$; |
|
102 |
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
|
103 |
||
104 |
\medskip |
|
109 | 105 |
\nn{ |
106 |
%At the moment I'm a little confused about orientations, and more specifically |
|
107 |
%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
|
108 |
Maybe need a discussion about what the boundary of a manifold with a |
|
109 |
structure (e.g. orientation) means. |
|
94 | 110 |
Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
111 |
Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
|
112 |
first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
|
113 |
equipped with an orientation of its once-stabilized tangent bundle. |
|
114 |
Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
|
115 |
their $k$ times stabilized tangent bundles. |
|
115 | 116 |
Probably should also have a framing of the stabilized dimensions in order to indicate which |
117 |
side the bounded manifold is on. |
|
94 | 118 |
For the moment just stick with unoriented manifolds.} |
119 |
\medskip |
|
120 |
||
121 |
We have just argued that the boundary of a morphism has no preferred splitting into |
|
122 |
domain and range, but the converse meets with our approval. |
|
123 |
That is, given compatible domain and range, we should be able to combine them into |
|
124 |
the full boundary of a morphism: |
|
125 |
||
126 |
\xxpar{Domain $+$ range $\to$ boundary:} |
|
103 | 127 |
{Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$), |
128 |
$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere. |
|
94 | 129 |
Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
130 |
two maps $\bd: \cC(B_i)\to \cC(E)$. |
|
131 |
Then (axiom) we have an injective map |
|
132 |
\[ |
|
133 |
\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
|
134 |
\] |
|
135 |
which is natural with respect to the actions of homeomorphisms.} |
|
136 |
||
137 |
Note that we insist on injectivity above. |
|
109 | 138 |
|
94 | 139 |
Let $\cC(S)_E$ denote the image of $\gl_E$. |
109 | 140 |
We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
141 |
||
110 | 142 |
We will call the projection $\cC(S)_E \to \cC(B_i)$ |
143 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
|
144 |
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
|
145 |
These restriction maps can be thought of as |
|
94 | 146 |
domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
147 |
||
103 | 148 |
If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
94 | 149 |
as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
150 |
||
151 |
Next we consider composition of morphisms. |
|
152 |
For $n$-categories which lack strong duality, one usually considers |
|
153 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
154 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
155 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
156 |
one general type of composition which can be in any ``direction". |
|
157 |
||
158 |
\xxpar{Composition:} |
|
103 | 159 |
{Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
160 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball. |
|
161 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
94 | 162 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
163 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
164 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
165 |
Then (axiom) we have a map |
|
166 |
\[ |
|
167 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
168 |
\] |
|
169 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
170 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
171 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
172 |
(For $k=n$, see below.)} |
|
173 |
||
95 | 174 |
\xxpar{Strict associativity:} |
102 | 175 |
{The composition (gluing) maps above are strictly associative.} |
176 |
||
109 | 177 |
Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
110 | 178 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
179 |
a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
|
180 |
Compositions of boundary and restriction maps will also be called restriction maps. |
|
181 |
For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
182 |
restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
183 |
||
184 |
%More notation and terminology: |
|
185 |
%We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction} |
|
186 |
%map |
|
109 | 187 |
|
102 | 188 |
The above two axioms are equivalent to the following axiom, |
189 |
which we state in slightly vague form. |
|
190 |
||
191 |
\xxpar{Multi-composition:} |
|
192 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
193 |
into small $k$-balls, there is a |
|
194 |
map from an appropriate subset (like a fibered product) |
|
195 |
of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
|
95 | 196 |
and these various $m$-fold composition maps satisfy an |
102 | 197 |
operad-type strict associativity condition.} |
95 | 198 |
|
199 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
200 |
||
201 |
\xxpar{Product (identity) morphisms:} |
|
103 | 202 |
{Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |
95 | 203 |
Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
204 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
|
205 |
\[ \xymatrix{ |
|
96 | 206 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
95 | 207 |
X \ar[r]^{f} & X' |
208 |
} \] |
|
109 | 209 |
commutes, then we have |
210 |
\[ |
|
211 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
212 |
\] |
|
213 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
214 |
\[ |
|
215 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
216 |
\] |
|
217 |
and |
|
218 |
\[ |
|
219 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
220 |
\] |
|
122 | 221 |
\nn{if pinched boundary, then remove first case above} |
109 | 222 |
Product morphisms are associative: |
223 |
\[ |
|
224 |
(a\times D)\times D' = a\times (D\times D') . |
|
225 |
\] |
|
226 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
227 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
110 | 228 |
Product morphisms are compatible with restriction: |
229 |
\[ |
|
230 |
\res_{X\times E}(a\times D) = a\times E |
|
231 |
\] |
|
232 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
109 | 233 |
} |
95 | 234 |
|
110 | 235 |
\nn{need even more subaxioms for product morphisms?} |
95 | 236 |
|
122 | 237 |
\nn{Almost certainly we need a little more than the above axiom. |
238 |
More specifically, in order to bootstrap our way from the top dimension |
|
239 |
properties of identity morphisms to low dimensions, we need regular products, |
|
240 |
pinched products and even half-pinched products. |
|
241 |
I'm not sure what the best way to cleanly axiomatize the properties of these various is. |
|
242 |
For the moment, I'll assume that all flavors of the product are at |
|
243 |
our disposal, and I'll plan on revising the axioms later.} |
|
244 |
||
128 | 245 |
\nn{current idea for fixing this: make the above axiom a ``preliminary version" |
246 |
(as we have already done with some of the other axioms), then state the official |
|
247 |
axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
|
248 |
one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
|
249 |
the alternative is to give some sort of local criterion for what's allowed. |
|
250 |
state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
|
251 |
} |
|
252 |
||
95 | 253 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
254 |
The last axiom (below), concerning actions of |
|
255 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
256 |
||
257 |
We start with the plain $n$-category case. |
|
258 |
||
259 |
\xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
|
103 | 260 |
{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
95 | 261 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
96 | 262 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.} |
263 |
||
264 |
We will strengthen the above axiom in two ways. |
|
265 |
(Amusingly, these two ways are related to each of the two senses of the term |
|
266 |
``pseudo-isotopy".) |
|
267 |
||
268 |
First, we require that $f$ act trivially on $\cC(X)$ if it is pseudo-isotopic to the identity |
|
269 |
in the sense of homeomorphisms of mapping cylinders. |
|
270 |
This is motivated by TQFT considerations: |
|
271 |
If the mapping cylinder of $f$ is homeomorphic to the mapping cylinder of the identity, |
|
272 |
then these two $n{+}1$-manifolds should induce the same map from $\cC(X)$ to itself. |
|
273 |
\nn{is there a non-TQFT reason to require this?} |
|
94 | 274 |
|
96 | 275 |
Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity. |
103 | 276 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 277 |
Let $J$ be a 1-ball (interval). |
278 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 279 |
(Here we use the ``pinched" version of $Y\times J$. |
280 |
\nn{need notation for this}) |
|
96 | 281 |
We define a map |
282 |
\begin{eqnarray*} |
|
283 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
284 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
285 |
\end{eqnarray*} |
|
286 |
\nn{need to say something somewhere about pinched boundary convention for products} |
|
287 |
We will call $\psi_{Y,J}$ an extended isotopy. |
|
97 | 288 |
\nn{or extended homeomorphism? see below.} |
289 |
\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
|
290 |
extended isotopies are also plain isotopies, so |
|
291 |
no extension necessary} |
|
96 | 292 |
It can be thought of as the action of the inverse of |
293 |
a map which projects a collar neighborhood of $Y$ onto $Y$. |
|
294 |
(This sort of collapse map is the other sense of ``pseudo-isotopy".) |
|
295 |
\nn{need to check this} |
|
296 |
||
297 |
The revised axiom is |
|
298 |
||
299 |
\xxpar{Pseudo and extended isotopy invariance in dimension $n$:} |
|
103 | 300 |
{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
96 | 301 |
to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
302 |
Then $f$ acts trivially on $\cC(X)$.} |
|
303 |
||
304 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
94 | 305 |
|
97 | 306 |
\smallskip |
307 |
||
308 |
For $A_\infty$ $n$-categories, we replace |
|
309 |
isotopy invariance with the requirement that families of homeomorphisms act. |
|
310 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
|
311 |
||
312 |
\xxpar{Families of homeomorphisms act.} |
|
313 |
{For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
|
314 |
\[ |
|
315 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
316 |
\] |
|
317 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
|
318 |
which fix $\bd X$. |
|
319 |
These action maps are required to be associative up to homotopy |
|
320 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
321 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
322 |
\nn{repeat diagram here?} |
|
323 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} |
|
324 |
||
325 |
We should strengthen the above axiom to apply to families of extended homeomorphisms. |
|
109 | 326 |
To do this we need to explain how extended homeomorphisms form a topological space. |
97 | 327 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
328 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
|
329 |
\nn{need to also say something about collaring homeomorphisms.} |
|
330 |
\nn{this paragraph needs work.} |
|
331 |
||
103 | 332 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
333 |
into a plain $n$-category (enriched over graded groups). |
|
97 | 334 |
\nn{say more here?} |
335 |
In the other direction, if we enrich over topological spaces instead of chain complexes, |
|
336 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
|
337 |
instead of $C_*(\Homeo_\bd(X))$. |
|
338 |
Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
|
339 |
type $A_\infty$ $n$-category. |
|
340 |
||
99 | 341 |
\medskip |
97 | 342 |
|
99 | 343 |
The alert reader will have already noticed that our definition of (plain) $n$-category |
344 |
is extremely similar to our definition of topological fields. |
|
345 |
The only difference is that for the $n$-category definition we restrict our attention to balls |
|
346 |
(and their boundaries), while for fields we consider all manifolds. |
|
347 |
\nn{also: difference at the top dimension; fix this} |
|
348 |
Thus a system of fields determines an $n$-category simply by restricting our attention to |
|
349 |
balls. |
|
350 |
The $n$-category can be thought of as the local part of the fields. |
|
351 |
Conversely, given an $n$-category we can construct a system of fields via |
|
352 |
\nn{gluing, or a universal construction} |
|
109 | 353 |
\nn{see subsection below} |
99 | 354 |
|
355 |
\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
|
356 |
of fields. |
|
357 |
The universal (colimit) construction becomes our generalized definition of blob homology. |
|
358 |
Need to explain how it relates to the old definition.} |
|
97 | 359 |
|
95 | 360 |
\medskip |
361 |
||
101 | 362 |
\nn{these examples need to be fleshed out a bit more} |
363 |
||
364 |
Examples of plain $n$-categories: |
|
365 |
\begin{itemize} |
|
366 |
||
367 |
\item Let $F$ be a closed $m$-manifold (e.g.\ a point). |
|
368 |
Let $T$ be a topological space. |
|
369 |
For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
|
370 |
all maps from $X\times F$ to $T$. |
|
371 |
For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
|
103 | 372 |
homotopies fixed on $\bd X \times F$. |
101 | 373 |
(Note that homotopy invariance implies isotopy invariance.) |
374 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
375 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
376 |
||
377 |
\item We can linearize the above example as follows. |
|
378 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
|
379 |
(e.g.\ the trivial cocycle). |
|
380 |
For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
|
381 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
|
382 |
the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
|
383 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
|
384 |
$h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
|
385 |
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
|
386 |
||
387 |
\item Given a traditional $n$-category $C$ (with strong duality etc.), |
|
388 |
define $\cC(X)$ (with $\dim(X) < n$) |
|
389 |
to be the set of all $C$-labeled sub cell complexes of $X$. |
|
390 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
|
391 |
combinations of $C$-labeled sub cell complexes of $X$ |
|
392 |
modulo the kernel of the evaluation map. |
|
393 |
Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
|
394 |
and with the same labeling as $a$. |
|
102 | 395 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
396 |
Define $\cC(X)$, for $\dim(X) < n$, |
|
397 |
to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
|
398 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
|
399 |
to be the dual Hilbert space $A(X\times F; c)$. |
|
101 | 400 |
\nn{refer elsewhere for details?} |
401 |
||
402 |
\item Variation on the above examples: |
|
103 | 403 |
We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
101 | 404 |
for example product boundary conditions or take the union over all boundary conditions. |
102 | 405 |
\nn{maybe should not emphasize this case, since it's ``better" in some sense |
406 |
to think of these guys as affording a representation |
|
407 |
of the $n{+}1$-category associated to $\bd F$.} |
|
101 | 408 |
|
125 | 409 |
\item \nn{should add bordism $n$-cat} |
410 |
||
101 | 411 |
\end{itemize} |
412 |
||
413 |
||
414 |
Examples of $A_\infty$ $n$-categories: |
|
415 |
\begin{itemize} |
|
416 |
||
417 |
\item Same as in example \nn{xxxx} above (fiber $F$, target space $T$), |
|
418 |
but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex |
|
419 |
$C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
|
420 |
and $C_*$ denotes singular chains. |
|
421 |
||
422 |
\item |
|
423 |
Given a plain $n$-category $C$, |
|
424 |
define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball |
|
425 |
and $\bc^C_*$ denotes the blob complex based on $C$. |
|
426 |
||
125 | 427 |
\item \nn{should add $\infty$ version of bordism $n$-cat} |
428 |
||
101 | 429 |
\end{itemize} |
95 | 430 |
|
108 | 431 |
|
432 |
||
433 |
||
434 |
||
435 |
||
436 |
\subsection{From $n$-categories to systems of fields} |
|
113 | 437 |
\label{ss:ncat_fields} |
108 | 438 |
|
439 |
We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
|
440 |
||
441 |
Let $W$ be a $k$-manifold, $1\le k \le n$. |
|
442 |
We will define a set $\cC(W)$. |
|
443 |
(If $k = n$ and our $k$-categories are enriched, then |
|
444 |
$\cC(W)$ will have additional structure; see below.) |
|
445 |
$\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
|
446 |
which we define next. |
|
447 |
||
448 |
Define a permissible decomposition of $W$ to be a decomposition |
|
449 |
\[ |
|
450 |
W = \bigcup_a X_a , |
|
451 |
\] |
|
452 |
where each $X_a$ is a $k$-ball. |
|
453 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
|
454 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
455 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
456 |
(The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
|
119 | 457 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
458 |
See Figure \ref{partofJfig}.) |
|
459 |
||
460 |
\begin{figure}[!ht] |
|
461 |
\begin{equation*} |
|
462 |
\mathfig{.63}{tempkw/zz2} |
|
463 |
\end{equation*} |
|
464 |
\caption{A small part of $\cJ(W)$} |
|
465 |
\label{partofJfig} |
|
466 |
\end{figure} |
|
467 |
||
108 | 468 |
|
469 |
$\cC$ determines |
|
470 |
a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
|
471 |
(possibly with additional structure if $k=n$). |
|
472 |
For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
|
473 |
\[ |
|
474 |
\psi_\cC(x) \sub \prod_a \cC(X_a) |
|
475 |
\] |
|
476 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
477 |
$X_a$ all agree. |
|
478 |
(Think fibered product.) |
|
479 |
If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
|
480 |
via the composition maps of $\cC$. |
|
112 | 481 |
(If $\dim(W) = n$ then we need to also make use of the monoidal |
482 |
product in the enriching category. |
|
483 |
\nn{should probably be more explicit here}) |
|
108 | 484 |
|
485 |
Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
|
112 | 486 |
In the plain (non-$A_\infty$) case, this means that |
487 |
for each decomposition $x$ there is a map |
|
108 | 488 |
$\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
489 |
above, and $\cC(W)$ is universal with respect to these properties. |
|
112 | 490 |
In the $A_\infty$ case, it means |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
491 |
\nn{.... need to check if there is a def in the literature before writing this down; |
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
492 |
homotopy colimit I think} |
112 | 493 |
|
494 |
More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
|
495 |
\[ |
|
496 |
\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
|
497 |
\] |
|
498 |
where $K$ is generated by all things of the form $a - g(a)$, where |
|
499 |
$a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
|
500 |
\to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
|
111 | 501 |
|
112 | 502 |
In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
503 |
is as follows. |
|
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
504 |
\nn{should probably rewrite this to be compatible with some standard reference} |
113 | 505 |
Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
112 | 506 |
Such sequences (for all $m$) form a simplicial set. |
507 |
Let |
|
508 |
\[ |
|
509 |
V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
|
510 |
\] |
|
113 | 511 |
where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$. |
112 | 512 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
513 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
|
514 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
515 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
|
516 |
\[ |
|
517 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) , |
|
518 |
\] |
|
519 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
|
520 |
is the usual map. |
|
521 |
\nn{need to say this better} |
|
522 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
523 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
524 |
of $A_\infty$ category} |
|
108 | 525 |
|
113 | 526 |
We will call $m$ the filtration degree of the complex. |
527 |
We can think of this construction as starting with a disjoint copy of a complex for each |
|
528 |
permissible decomposition (filtration degree 0). |
|
529 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
530 |
(filtration degree 1). |
|
531 |
Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2). |
|
532 |
And so on. |
|
533 |
||
108 | 534 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
535 |
||
536 |
It is easy to see that |
|
537 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
|
538 |
comprise a natural transformation of functors. |
|
539 |
||
540 |
\nn{need to finish explaining why we have a system of fields; |
|
541 |
need to say more about ``homological" fields? |
|
542 |
(actions of homeomorphisms); |
|
543 |
define $k$-cat $\cC(\cdot\times W)$} |
|
544 |
||
545 |
||
546 |
||
547 |
\subsection{Modules} |
|
95 | 548 |
|
101 | 549 |
Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
550 |
a.k.a.\ actions). |
|
102 | 551 |
The definition will be very similar to that of $n$-categories. |
109 | 552 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
110 | 553 |
\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
102 | 554 |
|
104 | 555 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 556 |
in the context of an $m{+}1$-dimensional TQFT. |
557 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
558 |
This will be explained in more detail as we present the axioms. |
|
559 |
||
560 |
Fix an $n$-category $\cC$. |
|
561 |
||
562 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
563 |
(standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
|
564 |
We call $B$ the ball and $N$ the marking. |
|
565 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
566 |
restricts to a homeomorphism of markings. |
|
567 |
||
568 |
\xxpar{Module morphisms} |
|
569 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
|
570 |
the category of marked $k$-balls and |
|
571 |
homeomorphisms to the category of sets and bijections.} |
|
572 |
||
573 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
574 |
||
104 | 575 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
576 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
577 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
578 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
579 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
580 |
(The union is along $N\times \bd W$.) |
|
110 | 581 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
582 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 583 |
|
103 | 584 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
585 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 586 |
|
587 |
\xxpar{Module boundaries, part 1:} |
|
588 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
|
104 | 589 |
the category of marked $k$-hemispheres and |
102 | 590 |
homeomorphisms to the category of sets and bijections.} |
591 |
||
104 | 592 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
593 |
||
102 | 594 |
\xxpar{Module boundaries, part 2:} |
595 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
|
596 |
These maps, for various $M$, comprise a natural transformation of functors.} |
|
597 |
||
110 | 598 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 599 |
|
600 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
601 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
602 |
and $c\in \cC(\bd M)$. |
|
603 |
||
604 |
\xxpar{Module domain $+$ range $\to$ boundary:} |
|
605 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
|
104 | 606 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
607 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
|
608 |
two maps $\bd: \cM(M_i)\to \cM(E)$. |
|
102 | 609 |
Then (axiom) we have an injective map |
610 |
\[ |
|
611 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
|
612 |
\] |
|
613 |
which is natural with respect to the actions of homeomorphisms.} |
|
614 |
||
110 | 615 |
Let $\cM(H)_E$ denote the image of $\gl_E$. |
616 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
|
617 |
||
618 |
||
103 | 619 |
\xxpar{Axiom yet to be named:} |
620 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
|
621 |
$\cM(H)\to \cC(H)$. |
|
622 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
|
623 |
These maps comprise a natural transformation of functors.} |
|
102 | 624 |
|
103 | 625 |
Note that combining the various boundary and restriction maps above |
110 | 626 |
(for both modules and $n$-categories) |
103 | 627 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
628 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 629 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
630 |
cutting submanifolds). |
|
103 | 631 |
This fact will be used below. |
102 | 632 |
|
104 | 633 |
In our example, the various restriction and gluing maps above come from |
634 |
restricting and gluing maps into $T$. |
|
635 |
||
636 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 637 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 638 |
(See Figure \ref{zzz3}.) |
103 | 639 |
|
119 | 640 |
\begin{figure}[!ht] |
641 |
\begin{equation*} |
|
642 |
\mathfig{.63}{tempkw/zz3} |
|
643 |
\end{equation*} |
|
644 |
\caption{Module composition (top); $n$-category action (bottom)} |
|
645 |
\label{zzz3} |
|
646 |
\end{figure} |
|
647 |
||
648 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 649 |
|
650 |
\xxpar{Module composition:} |
|
651 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
|
652 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
|
653 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
654 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
655 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
656 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
657 |
Then (axiom) we have a map |
|
658 |
\[ |
|
659 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
660 |
\] |
|
661 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
662 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
663 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
664 |
(For $k=n$, see below.)} |
|
665 |
||
119 | 666 |
|
667 |
||
103 | 668 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
669 |
module morphism. |
|
670 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
671 |
||
672 |
\xxpar{$n$-category action:} |
|
673 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
|
674 |
$X$ is a plain $k$-ball, |
|
675 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
676 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
677 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
678 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
679 |
Then (axiom) we have a map |
|
680 |
\[ |
|
681 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
682 |
\] |
|
683 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
684 |
to the intersection of the boundaries of $X$ and $M'$. |
|
685 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
686 |
(For $k=n$, see below.)} |
|
687 |
||
688 |
\xxpar{Module strict associativity:} |
|
689 |
{The composition and action maps above are strictly associative.} |
|
690 |
||
110 | 691 |
Note that the above associativity axiom applies to mixtures of module composition, |
692 |
action maps and $n$-category composition. |
|
119 | 693 |
See Figure \ref{zzz1b}. |
694 |
||
695 |
\begin{figure}[!ht] |
|
696 |
\begin{equation*} |
|
697 |
\mathfig{1}{tempkw/zz1b} |
|
698 |
\end{equation*} |
|
699 |
\caption{Two examples of mixed associativity} |
|
700 |
\label{zzz1b} |
|
701 |
\end{figure} |
|
702 |
||
110 | 703 |
|
704 |
The above three axioms are equivalent to the following axiom, |
|
103 | 705 |
which we state in slightly vague form. |
706 |
\nn{need figure for this} |
|
707 |
||
708 |
\xxpar{Module multi-composition:} |
|
709 |
{Given any decomposition |
|
710 |
\[ |
|
711 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
712 |
\] |
|
713 |
of a marked $k$-ball $M$ |
|
714 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
715 |
map from an appropriate subset (like a fibered product) |
|
716 |
of |
|
717 |
\[ |
|
718 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
719 |
\] |
|
720 |
to $\cM(M)$, |
|
721 |
and these various multifold composition maps satisfy an |
|
722 |
operad-type strict associativity condition.} |
|
723 |
||
724 |
(The above operad-like structure is analogous to the swiss cheese operad |
|
725 |
\nn{need citation}.) |
|
726 |
\nn{need to double-check that this is true.} |
|
727 |
||
728 |
\xxpar{Module product (identity) morphisms:} |
|
729 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
|
730 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
|
731 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
|
732 |
\[ \xymatrix{ |
|
733 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
|
734 |
M \ar[r]^{f} & M' |
|
735 |
} \] |
|
736 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
737 |
||
111 | 738 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
103 | 739 |
|
110 | 740 |
\nn{** marker --- resume revising here **} |
741 |
||
103 | 742 |
There are two alternatives for the next axiom, according whether we are defining |
743 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
744 |
In the plain case we require |
|
745 |
||
746 |
\xxpar{Pseudo and extended isotopy invariance in dimension $n$:} |
|
747 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
|
748 |
to the identity on $\bd M$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
|
749 |
Then $f$ acts trivially on $\cM(M)$.} |
|
750 |
||
751 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
752 |
||
753 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
754 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
755 |
on $\bd B \setmin N$. |
|
756 |
||
757 |
For $A_\infty$ modules we require |
|
758 |
||
759 |
\xxpar{Families of homeomorphisms act.} |
|
760 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
|
761 |
\[ |
|
762 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
763 |
\] |
|
764 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
765 |
which fix $\bd M$. |
|
766 |
These action maps are required to be associative up to homotopy |
|
767 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
768 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
769 |
\nn{repeat diagram here?} |
|
770 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
771 |
||
772 |
\medskip |
|
102 | 773 |
|
104 | 774 |
Note that the above axioms imply that an $n$-category module has the structure |
775 |
of an $n{-}1$-category. |
|
776 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
777 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
778 |
above the non-marked boundary component of $J$. |
|
779 |
\nn{give figure for this, or say more?} |
|
780 |
Then $\cE$ has the structure of an $n{-}1$-category. |
|
102 | 781 |
|
105 | 782 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
783 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
784 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
785 |
of marked 1-balls, call them left-marked and right-marked, |
|
786 |
and hence there are two types of modules, call them right modules and left modules. |
|
787 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
788 |
there is no left/right module distinction. |
|
789 |
||
130 | 790 |
\medskip |
791 |
||
792 |
Examples of modules: |
|
793 |
\begin{itemize} |
|
794 |
\item |
|
795 |
\end{itemize} |
|
108 | 796 |
|
797 |
\subsection{Modules as boundary labels} |
|
112 | 798 |
\label{moddecss} |
108 | 799 |
|
800 |
Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
801 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
802 |
component $\bd_i W$ of $W$. |
|
803 |
||
804 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
|
805 |
\nn{give ref} |
|
806 |
(If $k = n$ and our $k$-categories are enriched, then |
|
807 |
$\cC(W, \cN)$ will have additional structure; see below.) |
|
808 |
||
809 |
Define a permissible decomposition of $W$ to be a decomposition |
|
810 |
\[ |
|
811 |
W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
|
812 |
\] |
|
813 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
814 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
815 |
with $M_{ib}\cap\bd_i W$ being the marking. |
|
816 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
|
817 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
818 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
819 |
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
|
820 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
|
821 |
||
822 |
$\cN$ determines |
|
823 |
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
|
824 |
(possibly with additional structure if $k=n$). |
|
825 |
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
|
826 |
\[ |
|
111 | 827 |
\psi_\cN(x) \sub (\prod_a \cC(X_a)) \times (\prod_{ib} \cN_i(M_{ib})) |
108 | 828 |
\] |
829 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
830 |
$X_a$ and $M_{ib}$ all agree. |
|
831 |
(Think fibered product.) |
|
832 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
|
833 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
834 |
||
835 |
Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
|
836 |
In other words, for each decomposition $x$ there is a map |
|
837 |
$\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps |
|
838 |
above, and $\cC(W, \cN)$ is universal with respect to these properties. |
|
839 |
||
112 | 840 |
More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$. |
841 |
\nn{need to say more?} |
|
842 |
||
108 | 843 |
\nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.} |
844 |
||
845 |
\subsection{Tensor products} |
|
105 | 846 |
|
112 | 847 |
Next we consider tensor products. |
848 |
||
849 |
\nn{what about self tensor products /coends ?} |
|
105 | 850 |
|
108 | 851 |
\nn{maybe ``tensor product" is not the best name?} |
852 |
||
106 | 853 |
\nn{start with (less general) tensor products; maybe change this later} |
105 | 854 |
|
108 | 855 |
|
107 | 856 |
Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
857 |
(If $k=1$ and manifolds are oriented, then one should be |
|
858 |
a left module and the other a right module.) |
|
109 | 859 |
We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially) |
107 | 860 |
on a choice of 1-ball (interval) $J$. |
861 |
||
112 | 862 |
Let $p$ and $p'$ be the boundary points of $J$. |
863 |
Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with |
|
864 |
$X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}. |
|
865 |
Let |
|
106 | 866 |
\[ |
112 | 867 |
\cT(X) \deq \cC(X\times J, \cM, \cM') , |
106 | 868 |
\] |
112 | 869 |
where the right hand side is the colimit construction defined in Subsection \ref{moddecss}. |
870 |
It is not hard to see that $\cT$ becomes an $n{-}1$-category. |
|
871 |
\nn{maybe follows from stuff (not yet written) in previous subsection?} |
|
106 | 872 |
|
107 | 873 |
|
874 |
||
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
875 |
\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
|
876 |
|
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
877 |
Outline: |
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
878 |
\begin{itemize} |
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
879 |
\item |
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
115
diff
changeset
|
880 |
\end{itemize} |
107 | 881 |
|
882 |
||
101 | 883 |
|
884 |
\medskip |
|
885 |
\hrule |
|
886 |
\medskip |
|
887 |
||
95 | 888 |
\nn{to be continued...} |
101 | 889 |
\medskip |
98 | 890 |
|
891 |
||
892 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
|
893 |
a separate paper): |
|
894 |
\begin{itemize} |
|
895 |
\item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
|
896 |
\item conversely, our def implies other defs |
|
105 | 897 |
\item do same for modules; maybe an appendix on relating topological |
898 |
vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
|
98 | 899 |
\item traditional $A_\infty$ 1-cat def implies our def |
99 | 900 |
\item ... and vice-versa (already done in appendix) |
98 | 901 |
\item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
902 |
\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
|
903 |
\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
|
904 |
a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
130 | 905 |
\item morphisms of modules; show that it's adjoint to tensor product |
98 | 906 |
\end{itemize} |
907 |
||
134 | 908 |
\nn{Some salvaged paragraphs that we might want to work back in:} |
909 |
\hrule |
|
98 | 910 |
|
134 | 911 |
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.) |
912 |
||
913 |
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 |
|
914 |
\begin{align*} |
|
915 |
\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)), |
|
916 |
\end{align*} |
|
917 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
|
918 |
||
919 |
We now give two motivating examples, as theorems constructing other homological systems of fields, |
|
920 |
||
921 |
||
922 |
\begin{thm} |
|
923 |
For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as |
|
924 |
\begin{equation*} |
|
925 |
\Xi(M) = \CM{M}{X}. |
|
926 |
\end{equation*} |
|
927 |
\end{thm} |
|
928 |
||
929 |
\begin{thm} |
|
930 |
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 |
|
931 |
\begin{equation*} |
|
932 |
\cF^{\times F}(M) = \cB_*(M \times F, \cF). |
|
933 |
\end{equation*} |
|
934 |
\end{thm} |
|
935 |
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. |
|
936 |
||
937 |
||
938 |
In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields. |
|
939 |
||
940 |
||
941 |
\begin{thm} |
|
942 |
\begin{equation*} |
|
943 |
\cB_*(M, \Xi) \iso \Xi(M) |
|
944 |
\end{equation*} |
|
945 |
\end{thm} |
|
946 |
||
947 |
\begin{thm}[Product formula] |
|
948 |
Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields, |
|
949 |
there is a quasi-isomorphism |
|
950 |
\begin{align*} |
|
951 |
\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) |
|
952 |
\end{align*} |
|
953 |
\end{thm} |
|
954 |
||
955 |
\begin{question} |
|
956 |
Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
|
957 |
\end{question} |
|
958 |
||
959 |
\hrule |