84 boundary of a morphism. |
84 boundary of a morphism. |
85 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
85 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
86 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
86 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
87 $1\le k \le n$. |
87 $1\le k \le n$. |
88 At first it might seem that we need another axiom for this, but in fact once we have |
88 At first it might seem that we need another axiom for this, but in fact once we have |
89 all the axioms in the subsection for $0$ through $k-1$ we can use a coend |
89 all the axioms in the subsection for $0$ through $k-1$ we can use a colimit |
90 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
90 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
91 to spheres (and any other manifolds): |
91 to spheres (and any other manifolds): |
92 |
92 |
93 \begin{prop} |
93 \begin{prop} |
94 \label{axiom:spheres} |
94 \label{axiom:spheres} |
95 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
95 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
96 the category of $k{-}1$-spheres and |
96 the category of $k{-}1$-spheres and |
97 homeomorphisms to the category of sets and bijections. |
97 homeomorphisms to the category of sets and bijections. |
98 \end{prop} |
98 \end{prop} |
99 |
99 |
100 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
100 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
101 |
101 |
102 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
102 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
103 |
103 |
104 \begin{axiom}[Boundaries]\label{nca-boundary} |
104 \begin{axiom}[Boundaries]\label{nca-boundary} |
105 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
105 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
106 These maps, for various $X$, comprise a natural transformation of functors. |
106 These maps, for various $X$, comprise a natural transformation of functors. |
107 \end{axiom} |
107 \end{axiom} |
108 |
108 |
109 (Note that the first ``$\bd$" above is part of the data for the category, |
109 (Note that the first ``$\bd$" above is part of the data for the category, |
110 while the second is the ordinary boundary of manifolds.) |
110 while the second is the ordinary boundary of manifolds.) |
111 |
111 |
112 Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
112 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
113 |
113 |
114 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
114 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
115 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
115 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
116 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
116 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
117 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
117 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
118 and all the structure maps of the $n$-category should be compatible with the auxiliary |
118 and all the structure maps of the $n$-category should be compatible with the auxiliary |
119 category structure. |
119 category structure. |
120 Note that this auxiliary structure is only in dimension $n$; |
120 Note that this auxiliary structure is only in dimension $n$; |
121 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
121 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
140 |
140 |
141 We have just argued that the boundary of a morphism has no preferred splitting into |
141 We have just argued that the boundary of a morphism has no preferred splitting into |
142 domain and range, but the converse meets with our approval. |
142 domain and range, but the converse meets with our approval. |
143 That is, given compatible domain and range, we should be able to combine them into |
143 That is, given compatible domain and range, we should be able to combine them into |
144 the full boundary of a morphism. |
144 the full boundary of a morphism. |
145 The following proposition follows from the coend construction used to define $\cC_{k-1}$ |
145 The following proposition follows from the colimit construction used to define $\cl{\cC}_{k-1}$ |
146 on spheres. |
146 on spheres. |
147 |
147 |
148 \begin{prop}[Boundary from domain and range] |
148 \begin{prop}[Boundary from domain and range] |
149 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
149 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
150 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
150 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
151 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
151 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
152 two maps $\bd: \cC(B_i)\to \cC(E)$. |
152 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
153 Then we have an injective map |
153 Then we have an injective map |
154 \[ |
154 \[ |
155 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \into \cC(S) |
155 \gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
156 \] |
156 \] |
157 which is natural with respect to the actions of homeomorphisms. |
157 which is natural with respect to the actions of homeomorphisms. |
158 (When $k=1$ we stipulate that $\cC(E)$ is a point, so that the above fibered product |
158 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
159 becomes a normal product.) |
159 becomes a normal product.) |
160 \end{prop} |
160 \end{prop} |
161 |
161 |
162 \begin{figure}[!ht] |
162 \begin{figure}[!ht] |
163 $$ |
163 $$ |
164 \begin{tikzpicture}[%every label/.style={green} |
164 \begin{tikzpicture}[%every label/.style={green} |
165 ] |
165 ] |
166 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
166 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
167 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
167 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
168 \draw (S) arc (-90:90:1); |
168 \draw (S) arc (-90:90:1); |
169 \draw (N) arc (90:270:1); |
169 \draw (N) arc (90:270:1); |
170 \node[left] at (-1,1) {$B_1$}; |
170 \node[left] at (-1,1) {$B_1$}; |
173 $$ |
173 $$ |
174 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
174 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
175 |
175 |
176 Note that we insist on injectivity above. |
176 Note that we insist on injectivity above. |
177 |
177 |
178 Let $\cC(S)_E$ denote the image of $\gl_E$. |
178 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
179 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
179 We will refer to elements of $\\cl{cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
180 |
180 |
181 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
181 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
182 as above, then we define $\cC(X)_E = \bd^{-1}(\cC(\bd X)_E)$. |
182 as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
183 |
183 |
184 We will call the projection $\cC(S)_E \to \cC(B_i)$ |
184 We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
185 a {\it restriction} map and write $\res_{B_i}(a)$ |
185 a {\it restriction} map and write $\res_{B_i}(a)$ |
186 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
186 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
187 More generally, we also include under the rubric ``restriction map" the |
187 More generally, we also include under the rubric ``restriction map" the |
188 the boundary maps of Axiom \ref{nca-boundary} above, |
188 the boundary maps of Axiom \ref{nca-boundary} above, |
189 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
189 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
190 of restriction maps. |
190 of restriction maps. |
191 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
191 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |