1275 \end{module-axiom} |
1275 \end{module-axiom} |
1276 |
1276 |
1277 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1277 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1278 |
1278 |
1279 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1279 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1280 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
1280 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1281 and $c\in \cC(\bd M)$. |
|
1282 |
1281 |
1283 \begin{lem}[Boundary from domain and range] |
1282 \begin{lem}[Boundary from domain and range] |
1284 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1283 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1285 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1284 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1286 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1285 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1305 |
1304 |
1306 Note that combining the various boundary and restriction maps above |
1305 Note that combining the various boundary and restriction maps above |
1307 (for both modules and $n$-categories) |
1306 (for both modules and $n$-categories) |
1308 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1307 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1309 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
1308 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
1310 The subset is the subset of morphisms which are appropriately splittable (transverse to the |
1309 This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the |
1311 cutting submanifolds). |
1310 cutting submanifolds). |
1312 This fact will be used below. |
1311 This fact will be used below. |
1313 |
1312 |
1314 In our example, the various restriction and gluing maps above come from |
1313 In our example, the various restriction and gluing maps above come from |
1315 restricting and gluing maps into $T$. |
1314 restricting and gluing maps into $T$. |
1331 \begin{module-axiom}[Module composition] |
1330 \begin{module-axiom}[Module composition] |
1332 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
1331 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
1333 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1332 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1334 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
1333 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
1335 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
1334 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
1336 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
1335 We have restriction (domain or range) maps $\cM(M_i)\trans E \to \cM(Y)$. |
1337 Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
1336 Let $\cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E$ denote the fibered product of these two maps. |
1338 Then (axiom) we have a map |
1337 Then (axiom) we have a map |
1339 \[ |
1338 \[ |
1340 \gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
1339 \gl_Y : \cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E \to \cM(M) \trans E |
1341 \] |
1340 \] |
1342 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1341 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1343 to the intersection of the boundaries of $M$ and $M_i$. |
1342 to the intersection of the boundaries of $M$ and $M_i$. |
1344 If $k < n$, |
1343 If $k < n$, |
1345 or if $k=n$ and we are in the $A_\infty$ case, |
1344 or if $k=n$ and we are in the $A_\infty$ case, |
1355 \begin{module-axiom}[$n$-category action] |
1354 \begin{module-axiom}[$n$-category action] |
1356 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1355 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1357 $X$ is a plain $k$-ball, |
1356 $X$ is a plain $k$-ball, |
1358 and $Y = X\cap M'$ is a $k{-}1$-ball. |
1357 and $Y = X\cap M'$ is a $k{-}1$-ball. |
1359 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
1358 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
1360 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
1359 We have restriction maps $\cM(M') \trans E \to \cC(Y)$ and $\cC(X) \trans E\to \cC(Y)$. |
1361 Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
1360 Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M') \trans E$ denote the fibered product of these two maps. |
1362 Then (axiom) we have a map |
1361 Then (axiom) we have a map |
1363 \[ |
1362 \[ |
1364 \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
1363 \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')\trans E \to \cM(M) \trans E |
1365 \] |
1364 \] |
1366 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1365 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1367 to the intersection of the boundaries of $X$ and $M'$. |
1366 to the intersection of the boundaries of $X$ and $M'$. |
1368 If $k < n$, |
1367 If $k < n$, |
1369 or if $k=n$ and we are in the $A_\infty$ case, |
1368 or if $k=n$ and we are in the $A_\infty$ case, |