195 |
195 |
196 Note that we insist on injectivity above. |
196 Note that we insist on injectivity above. |
197 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
197 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
198 %\nn{we might want a more official looking proof...} |
198 %\nn{we might want a more official looking proof...} |
199 |
199 |
200 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
200 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
201 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
201 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
202 |
202 |
203 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
203 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
204 as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
204 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
205 |
205 |
206 We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
206 We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ |
207 a {\it restriction} map and write $\res_{B_i}(a)$ |
207 a {\it restriction} map and write $\res_{B_i}(a)$ |
208 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
208 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$. |
209 More generally, we also include under the rubric ``restriction map" |
209 More generally, we also include under the rubric ``restriction map" |
210 the boundary maps of Axiom \ref{nca-boundary} above, |
210 the boundary maps of Axiom \ref{nca-boundary} above, |
211 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
211 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
212 of restriction maps. |
212 of restriction maps. |
213 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
213 In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ |
214 ($i = 1, 2$, notation from previous paragraph). |
214 ($i = 1, 2$, notation from previous paragraph). |
215 These restriction maps can be thought of as |
215 These restriction maps can be thought of as |
216 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
216 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
217 |
217 |
218 |
218 |
227 \label{axiom:composition} |
227 \label{axiom:composition} |
228 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
228 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
229 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
229 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
230 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
230 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
231 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
231 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
232 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
232 We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$. |
233 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
233 Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. |
234 We have a map |
234 We have a map |
235 \[ |
235 \[ |
236 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
236 \gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)_E |
237 \] |
237 \] |
238 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
238 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
239 to the intersection of the boundaries of $B$ and $B_i$. |
239 to the intersection of the boundaries of $B$ and $B_i$. |
240 If $k < n$, |
240 If $k < n$, |
241 or if $k=n$ and we are in the $A_\infty$ case, |
241 or if $k=n$ and we are in the $A_\infty$ case, |
267 \begin{figure}[!ht] |
267 \begin{figure}[!ht] |
268 $$\mathfig{.65}{ncat/strict-associativity}$$ |
268 $$\mathfig{.65}{ncat/strict-associativity}$$ |
269 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
269 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
270 |
270 |
271 We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
271 We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
272 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
272 In the other direction, we will call the projection from $\cC(B)\trans E$ to $\cC(B_i)\trans E$ |
273 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
273 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)\trans E$. |
274 %Compositions of boundary and restriction maps will also be called restriction maps. |
274 %Compositions of boundary and restriction maps will also be called restriction maps. |
275 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
275 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
276 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
276 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
277 |
277 |
278 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
278 We will write $\cC(B)\trans Y$ for the image of $\gl_Y$ in $\cC(B)$. |
279 We will call elements of $\cC(B)_Y$ morphisms which are |
279 We will call elements of $\cC(B)\trans Y$ morphisms which are |
280 ``splittable along $Y$'' or ``transverse to $Y$''. |
280 ``splittable along $Y$'' or ``transverse to $Y$''. |
281 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
281 We have $\cC(B)\trans Y \sub \cC(B)\trans E \sub \cC(B)$. |
282 |
282 |
283 More generally, let $\alpha$ be a splitting of $X$ into smaller balls. |
283 More generally, let $\alpha$ be a splitting of $X$ into smaller balls. |
284 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
284 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
285 the smaller balls to $X$. |
285 the smaller balls to $X$. |
286 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
286 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
678 |
678 |
679 An $n$-category consists of the following data: |
679 An $n$-category consists of the following data: |
680 \begin{itemize} |
680 \begin{itemize} |
681 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
681 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
682 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
682 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
683 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}); |
683 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}); |
684 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
684 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
685 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$; |
685 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$; |
686 \item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
686 \item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
687 \end{itemize} |
687 \end{itemize} |
688 The above data must satisfy the following conditions: |
688 The above data must satisfy the following conditions: |
1264 \] |
1264 \] |
1265 which is natural with respect to the actions of homeomorphisms.} |
1265 which is natural with respect to the actions of homeomorphisms.} |
1266 \end{lem} |
1266 \end{lem} |
1267 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1267 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1268 |
1268 |
1269 Let $\cl\cM(H)_E$ denote the image of $\gl_E$. |
1269 Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
1270 We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
1270 We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1271 |
1271 |
1272 \begin{lem}[Module to category restrictions] |
1272 \begin{lem}[Module to category restrictions] |
1273 {For each marked $k$-hemisphere $H$ there is a restriction map |
1273 {For each marked $k$-hemisphere $H$ there is a restriction map |
1274 $\cl\cM(H)\to \cC(H)$. |
1274 $\cl\cM(H)\to \cC(H)$. |
1275 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1275 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1329 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1329 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1330 $X$ is a plain $k$-ball, |
1330 $X$ is a plain $k$-ball, |
1331 and $Y = X\cap M'$ is a $k{-}1$-ball. |
1331 and $Y = X\cap M'$ is a $k{-}1$-ball. |
1332 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
1332 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
1333 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
1333 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
1334 Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
1334 Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
1335 Then (axiom) we have a map |
1335 Then (axiom) we have a map |
1336 \[ |
1336 \[ |
1337 \gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
1337 \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
1338 \] |
1338 \] |
1339 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1339 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1340 to the intersection of the boundaries of $X$ and $M'$. |
1340 to the intersection of the boundaries of $X$ and $M'$. |
1341 If $k < n$, |
1341 If $k < n$, |
1342 or if $k=n$ and we are in the $A_\infty$ case, |
1342 or if $k=n$ and we are in the $A_\infty$ case, |