16 |
16 |
17 \subsection{Definition of $n$-categories} |
17 \subsection{Definition of $n$-categories} |
18 |
18 |
19 Before proceeding, we need more appropriate definitions of $n$-categories, |
19 Before proceeding, we need more appropriate definitions of $n$-categories, |
20 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
20 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
21 (As is the case throughout this paper, by ``$n$-category" we mean |
21 (As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of |
22 a weak $n$-category with strong duality.) |
22 a `weak' $n$-category with `strong duality'.) |
23 |
23 |
24 The definitions presented below tie the categories more closely to the topology |
24 The definitions presented below tie the categories more closely to the topology |
25 and avoid combinatorial questions about, for example, the minimal sufficient |
25 and avoid combinatorial questions about, for example, the minimal sufficient |
26 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
26 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
27 For examples of topological origin, it is typically easy to show that they |
27 For examples of topological origin, it is typically easy to show that they |
44 |
44 |
45 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
45 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
46 the standard $k$-ball. |
46 the standard $k$-ball. |
47 In other words, |
47 In other words, |
48 |
48 |
49 \xxpar{Morphisms (preliminary version):} |
49 \begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
50 {For any $k$-manifold $X$ homeomorphic |
50 For any $k$-manifold $X$ homeomorphic |
51 to the standard $k$-ball, we have a set of $k$-morphisms |
51 to the standard $k$-ball, we have a set of $k$-morphisms |
52 $\cC_k(X)$.} |
52 $\cC_k(X)$. |
|
53 \end{preliminary-axiom} |
53 |
54 |
54 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
55 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
55 standard $k$-ball. |
56 standard $k$-ball. |
56 We {\it do not} assume that it is equipped with a |
57 We {\it do not} assume that it is equipped with a |
57 preferred homeomorphism to the standard $k$-ball. |
58 preferred homeomorphism to the standard $k$-ball. |
62 the boundary), we want a corresponding |
63 the boundary), we want a corresponding |
63 bijection of sets $f:\cC(X)\to \cC(Y)$. |
64 bijection of sets $f:\cC(X)\to \cC(Y)$. |
64 (This will imply ``strong duality", among other things.) |
65 (This will imply ``strong duality", among other things.) |
65 So we replace the above with |
66 So we replace the above with |
66 |
67 |
67 \xxpar{Morphisms:} |
68 \begin{axiom}[Morphisms] |
68 %\xxpar{Axiom 1 -- Morphisms:} |
69 \label{axiom:morphisms} |
69 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
70 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
70 the category of $k$-balls and |
71 the category of $k$-balls and |
71 homeomorphisms to the category of sets and bijections.} |
72 homeomorphisms to the category of sets and bijections. |
|
73 \end{axiom} |
|
74 |
72 |
75 |
73 (Note: We usually omit the subscript $k$.) |
76 (Note: We usually omit the subscript $k$.) |
74 |
77 |
75 We are being deliberately vague about what flavor of manifolds we are considering. |
78 We are so far being deliberately vague about what flavor of manifolds we are considering. |
76 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
79 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
77 They could be topological or PL or smooth. |
80 They could be topological or PL or smooth. |
78 \nn{need to check whether this makes much difference --- see pseudo-isotopy below} |
81 \nn{need to check whether this makes much difference --- see pseudo-isotopy below} |
79 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
82 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
80 to be fussier about corners.) |
83 to be fussier about corners.) |
91 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
94 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
92 We prefer to combine the domain and range into a single entity which we call the |
95 We prefer to combine the domain and range into a single entity which we call the |
93 boundary of a morphism. |
96 boundary of a morphism. |
94 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
97 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
95 |
98 |
96 \xxpar{Boundaries (domain and range), part 1:} |
99 \begin{axiom}[Boundaries (spheres)] |
97 {For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
100 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
98 the category of $k$-spheres and |
101 the category of $k$-spheres and |
99 homeomorphisms to the category of sets and bijections.} |
102 homeomorphisms to the category of sets and bijections. |
|
103 \end{axiom} |
100 |
104 |
101 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
105 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
102 |
106 |
103 \xxpar{Boundaries, part 2:} |
107 \begin{axiom}[Boundaries (maps)] |
104 {For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
108 For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
105 These maps, for various $X$, comprise a natural transformation of functors.} |
109 These maps, for various $X$, comprise a natural transformation of functors. |
|
110 \end{axiom} |
106 |
111 |
107 (Note that the first ``$\bd$" above is part of the data for the category, |
112 (Note that the first ``$\bd$" above is part of the data for the category, |
108 while the second is the ordinary boundary of manifolds.) |
113 while the second is the ordinary boundary of manifolds.) |
109 |
114 |
110 Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
115 Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
139 We have just argued that the boundary of a morphism has no preferred splitting into |
144 We have just argued that the boundary of a morphism has no preferred splitting into |
140 domain and range, but the converse meets with our approval. |
145 domain and range, but the converse meets with our approval. |
141 That is, given compatible domain and range, we should be able to combine them into |
146 That is, given compatible domain and range, we should be able to combine them into |
142 the full boundary of a morphism: |
147 the full boundary of a morphism: |
143 |
148 |
144 \xxpar{Domain $+$ range $\to$ boundary:} |
149 \begin{axiom}[Boundary from domain and range] |
145 {Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$), |
150 Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$, |
146 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
151 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
147 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
152 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
148 two maps $\bd: \cC(B_i)\to \cC(E)$. |
153 two maps $\bd: \cC(B_i)\to \cC(E)$. |
149 Then (axiom) we have an injective map |
154 Then we have an injective map |
150 \[ |
155 \[ |
151 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
156 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
152 \] |
157 \] |
153 which is natural with respect to the actions of homeomorphisms.} |
158 which is natural with respect to the actions of homeomorphisms. |
|
159 \end{axiom} |
154 |
160 |
155 \begin{figure}[!ht] |
161 \begin{figure}[!ht] |
156 $$ |
162 $$ |
157 \begin{tikzpicture}[every label/.style={green}] |
163 \begin{tikzpicture}[every label/.style={green}] |
158 \node[fill=black, circle, label=below:$E$](S) at (0,0) {}; |
164 \node[fill=black, circle, label=below:$E$](S) at (0,0) {}; |
185 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
191 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
186 (For example, vertical and horizontal composition of 2-morphisms.) |
192 (For example, vertical and horizontal composition of 2-morphisms.) |
187 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
193 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
188 one general type of composition which can be in any ``direction". |
194 one general type of composition which can be in any ``direction". |
189 |
195 |
190 \xxpar{Composition:} |
196 \begin{axiom}[Composition] |
191 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
197 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
192 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
198 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
193 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
199 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
194 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
200 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
195 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
201 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
196 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
202 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
199 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
205 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
200 \] |
206 \] |
201 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
207 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
202 to the intersection of the boundaries of $B$ and $B_i$. |
208 to the intersection of the boundaries of $B$ and $B_i$. |
203 If $k < n$ we require that $\gl_Y$ is injective. |
209 If $k < n$ we require that $\gl_Y$ is injective. |
204 (For $k=n$, see below.)} |
210 (For $k=n$, see below.) |
|
211 \end{axiom} |
205 |
212 |
206 \begin{figure}[!ht] |
213 \begin{figure}[!ht] |
207 $$\mathfig{.4}{tempkw/blah5}$$ |
214 $$\mathfig{.4}{tempkw/blah5}$$ |
208 \caption{From two balls to one ball}\label{blah5}\end{figure} |
215 \caption{From two balls to one ball}\label{blah5}\end{figure} |
209 |
216 |
210 \xxpar{Strict associativity:} |
217 \begin{axiom}[Strict associativity] |
211 {The composition (gluing) maps above are strictly associative.} |
218 The composition (gluing) maps above are strictly associative. |
|
219 \end{axiom} |
212 |
220 |
213 \begin{figure}[!ht] |
221 \begin{figure}[!ht] |
214 $$\mathfig{.65}{tempkw/blah6}$$ |
222 $$\mathfig{.65}{tempkw/blah6}$$ |
215 \caption{An example of strict associativity}\label{blah6}\end{figure} |
223 \caption{An example of strict associativity}\label{blah6}\end{figure} |
216 |
224 |
240 $$\mathfig{.8}{tempkw/blah7}$$ |
248 $$\mathfig{.8}{tempkw/blah7}$$ |
241 \caption{Operadish composition and associativity}\label{blah7}\end{figure} |
249 \caption{Operadish composition and associativity}\label{blah7}\end{figure} |
242 |
250 |
243 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
251 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
244 |
252 |
245 \xxpar{Product (identity) morphisms:} |
253 \begin{axiom}[Product (identity) morphisms] |
246 {Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |
254 Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |
247 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
255 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
248 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
256 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
249 \[ \xymatrix{ |
257 \[ \xymatrix{ |
250 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
258 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
251 X \ar[r]^{f} & X' |
259 X \ar[r]^{f} & X' |
272 Product morphisms are compatible with restriction: |
280 Product morphisms are compatible with restriction: |
273 \[ |
281 \[ |
274 \res_{X\times E}(a\times D) = a\times E |
282 \res_{X\times E}(a\times D) = a\times E |
275 \] |
283 \] |
276 for $E\sub \bd D$ and $a\in \cC(X)$. |
284 for $E\sub \bd D$ and $a\in \cC(X)$. |
277 } |
285 \end{axiom} |
278 |
286 |
279 \nn{need even more subaxioms for product morphisms?} |
287 \nn{need even more subaxioms for product morphisms?} |
280 |
288 |
281 \nn{Almost certainly we need a little more than the above axiom. |
289 \nn{Almost certainly we need a little more than the above axiom. |
282 More specifically, in order to bootstrap our way from the top dimension |
290 More specifically, in order to bootstrap our way from the top dimension |
299 The last axiom (below), concerning actions of |
307 The last axiom (below), concerning actions of |
300 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
308 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
301 |
309 |
302 We start with the plain $n$-category case. |
310 We start with the plain $n$-category case. |
303 |
311 |
304 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
312 \begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$} |
305 {Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
313 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
306 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
314 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
307 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.} |
315 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
|
316 \end{preliminary-axiom} |
308 |
317 |
309 This axiom needs to be strengthened to force product morphisms to act as the identity. |
318 This axiom needs to be strengthened to force product morphisms to act as the identity. |
310 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
319 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
311 Let $J$ be a 1-ball (interval). |
320 Let $J$ be a 1-ball (interval). |
312 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
321 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
331 It can be thought of as the action of the inverse of |
340 It can be thought of as the action of the inverse of |
332 a map which projects a collar neighborhood of $Y$ onto $Y$. |
341 a map which projects a collar neighborhood of $Y$ onto $Y$. |
333 |
342 |
334 The revised axiom is |
343 The revised axiom is |
335 |
344 |
336 \xxpar{Extended isotopy invariance in dimension $n$:} |
345 \begin{axiom}[Extended isotopy invariance in dimension $n$] |
337 {Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
346 \label{axiom:extended-isotopies} |
|
347 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
338 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
348 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
339 Then $f$ acts trivially on $\cC(X)$.} |
349 Then $f$ acts trivially on $\cC(X)$. |
|
350 \end{axiom} |
340 |
351 |
341 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
352 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
342 |
353 |
343 \smallskip |
354 \smallskip |
344 |
355 |
345 For $A_\infty$ $n$-categories, we replace |
356 For $A_\infty$ $n$-categories, we replace |
346 isotopy invariance with the requirement that families of homeomorphisms act. |
357 isotopy invariance with the requirement that families of homeomorphisms act. |
347 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
358 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
348 |
359 |
349 \xxpar{Families of homeomorphisms act in dimension $n$.} |
360 \begin{axiom}[Families of homeomorphisms act in dimension $n$] |
350 {For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
361 For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
351 \[ |
362 \[ |
352 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
363 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
353 \] |
364 \] |
354 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
365 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
355 which fix $\bd X$. |
366 which fix $\bd X$. |
356 These action maps are required to be associative up to homotopy |
367 These action maps are required to be associative up to homotopy |
357 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
368 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
358 a diagram like the one in Proposition \ref{CDprop} commutes. |
369 a diagram like the one in Proposition \ref{CDprop} commutes. |
359 \nn{repeat diagram here?} |
370 \nn{repeat diagram here?} |
360 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} |
371 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
|
372 \end{axiom} |
361 |
373 |
362 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
374 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
363 To do this we need to explain how extended homeomorphisms form a topological space. |
375 To do this we need to explain how extended homeomorphisms form a topological space. |
364 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
376 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
365 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
377 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |