31 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
31 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
32 |
32 |
33 \medskip |
33 \medskip |
34 |
34 |
35 The axioms for an $n$-category are spread throughout this section. |
35 The axioms for an $n$-category are spread throughout this section. |
36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, |
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37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and |
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38 \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace |
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39 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
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40 \nn{need to revise this after we're done rearranging the a-inf and enriched stuff} |
37 |
41 |
38 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
39 for $k{-}1$-morphisms. |
43 for $k{-}1$-morphisms. |
40 Readers who prefer things to be presented in a strictly logical order should read this subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
44 Readers who prefer things to be presented in a strictly logical order should read this |
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45 subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
41 |
46 |
42 \medskip |
47 \medskip |
43 |
48 |
44 There are many existing definitions of $n$-categories, with various intended uses. |
49 There are many existing definitions of $n$-categories, with various intended uses. |
45 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
50 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
50 and so on. |
55 and so on. |
51 (This allows for strict associativity; see \cite{ulrike-tillmann-2008,0909.2212}.) |
56 (This allows for strict associativity; see \cite{ulrike-tillmann-2008,0909.2212}.) |
52 Still other definitions (see, for example, \cite{MR2094071}) |
57 Still other definitions (see, for example, \cite{MR2094071}) |
53 model the $k$-morphisms on more complicated combinatorial polyhedra. |
58 model the $k$-morphisms on more complicated combinatorial polyhedra. |
54 |
59 |
55 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
60 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is |
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61 homeomorphic to the standard $k$-ball. |
56 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
62 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
57 to the standard $k$-ball. |
63 to the standard $k$-ball. |
58 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
64 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
59 standard $k$-ball. |
65 standard $k$-ball. |
60 We {\it do not} assume that it is equipped with a |
66 We {\it do not} assume that it is equipped with a |
138 \end{axiom} |
144 \end{axiom} |
139 |
145 |
140 Note that the first ``$\bd$" above is part of the data for the category, |
146 Note that the first ``$\bd$" above is part of the data for the category, |
141 while the second is the ordinary boundary of manifolds. |
147 while the second is the ordinary boundary of manifolds. |
142 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
148 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
143 |
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144 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
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145 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
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146 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
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147 with sufficient limits and colimits |
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148 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
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149 %\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} |
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150 and all the structure maps of the $n$-category should be compatible with the auxiliary |
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151 category structure. |
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152 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
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153 $\cC(Y; c)$ is just a plain set. |
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154 |
149 |
155 \medskip |
150 \medskip |
156 |
151 |
157 In order to simplify the exposition we have concentrated on the case of |
152 In order to simplify the exposition we have concentrated on the case of |
158 unoriented PL manifolds and avoided the question of what exactly we mean by |
153 unoriented PL manifolds and avoided the question of what exactly we mean by |
237 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
232 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
238 |
233 |
239 |
234 |
240 Next we consider composition of morphisms. |
235 Next we consider composition of morphisms. |
241 For $n$-categories which lack strong duality, one usually considers |
236 For $n$-categories which lack strong duality, one usually considers |
242 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
237 $k$ different types of composition of $k$-morphisms, each associated to a different ``direction". |
243 (For example, vertical and horizontal composition of 2-morphisms.) |
238 (For example, vertical and horizontal composition of 2-morphisms.) |
244 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
239 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
245 one general type of composition which can be in any ``direction". |
240 one general type of composition which can be in any direction. |
246 |
241 |
247 \begin{axiom}[Composition] |
242 \begin{axiom}[Composition] |
248 \label{axiom:composition} |
243 \label{axiom:composition} |
249 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
244 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
250 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
245 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
256 \[ |
251 \[ |
257 \gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)\trans E |
252 \gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)\trans E |
258 \] |
253 \] |
259 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
254 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
260 to the intersection of the boundaries of $B$ and $B_i$. |
255 to the intersection of the boundaries of $B$ and $B_i$. |
261 If $k < n$, |
256 If $k < n$ |
262 or if $k=n$ and we are in the $A_\infty$ case, |
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263 we require that $\gl_Y$ is injective. |
257 we require that $\gl_Y$ is injective. |
264 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.) |
258 %(For $k=n$ see below.) |
265 \end{axiom} |
259 \end{axiom} |
266 |
260 |
267 \begin{figure}[t] \centering |
261 \begin{figure}[t] \centering |
268 \begin{tikzpicture}[%every label/.style={green}, |
262 \begin{tikzpicture}[%every label/.style={green}, |
269 x=1.5cm,y=1.5cm] |
263 x=1.5cm,y=1.5cm] |
399 \end{tikzpicture} |
393 \end{tikzpicture} |
400 $$ |
394 $$ |
401 \caption{Examples of pinched products}\label{pinched_prods} |
395 \caption{Examples of pinched products}\label{pinched_prods} |
402 \end{figure} |
396 \end{figure} |
403 The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
397 The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
404 where we construct a traditional category from a disk-like category. |
398 where we construct a traditional 2-category from a disk-like 2-category. |
405 For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms |
399 For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms |
406 in 2-categories. |
400 in 2-categories. |
407 We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). |
401 We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). |
408 |
402 |
409 Define a {\it pinched product} to be a map |
403 Define a {\it pinched product} to be a map |
658 isotopy invariance with the requirement that families of homeomorphisms act. |
652 isotopy invariance with the requirement that families of homeomorphisms act. |
659 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
653 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
660 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
654 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
661 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
655 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
662 |
656 |
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657 \nn{begin temp relocation} |
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658 |
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659 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
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660 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
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661 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
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662 with sufficient limits and colimits |
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663 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
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664 %\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} |
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665 and all the structure maps of the $n$-category should be compatible with the auxiliary |
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666 category structure. |
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667 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
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668 $\cC(Y; c)$ is just a plain set. |
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669 |
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670 \nn{$k=n$ injectivity for a-inf (necessary?)} |
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671 or if $k=n$ and we are in the $A_\infty$ case, |
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672 |
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673 |
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674 \nn{end temp relocation} |
663 |
675 |
664 %\addtocounter{axiom}{-1} |
676 %\addtocounter{axiom}{-1} |
665 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
677 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
666 \label{axiom:families} |
678 \label{axiom:families} |
667 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
679 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |