876 The remaining data for the $A_\infty$ $n$-category |
876 The remaining data for the $A_\infty$ $n$-category |
877 --- composition and $\Diff(X\to X')$ action --- |
877 --- composition and $\Diff(X\to X')$ action --- |
878 also comes from the $\cE\cB_n$ action on $A$. |
878 also comes from the $\cE\cB_n$ action on $A$. |
879 \nn{should we spell this out?} |
879 \nn{should we spell this out?} |
880 |
880 |
881 \nn{Should remark that this is just Lurie's topological chiral homology construction |
881 \nn{Should remark that the associated hocolim for manifolds |
882 applied to $n$-balls (need to check that colims agree).} |
882 is agrees with Lurie's topological chiral homology construction; maybe wait |
|
883 until next subsection to say that?} |
883 |
884 |
884 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
885 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
885 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
886 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
886 an $\cE\cB_n$-algebra. |
887 an $\cE\cB_n$-algebra. |
887 \nn{The paper is already long; is it worth giving details here?} |
888 \nn{The paper is already long; is it worth giving details here?} |
888 \end{example} |
889 \end{example} |
889 |
890 |
890 |
891 |
891 |
|
892 |
|
893 |
|
894 |
|
895 %\subsection{From $n$-categories to systems of fields} |
|
896 \subsection{From balls to manifolds} |
892 \subsection{From balls to manifolds} |
897 \label{ss:ncat_fields} \label{ss:ncat-coend} |
893 \label{ss:ncat_fields} \label{ss:ncat-coend} |
898 In this section we describe how to extend an $n$-category $\cC$ as described above |
894 In this section we describe how to extend an $n$-category $\cC$ as described above |
899 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
895 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
900 This extension is a certain colimit, and we've chosen the notation to remind you of this. |
896 This extension is a certain colimit, and we've chosen the notation to remind you of this. |
1112 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1108 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1113 Call such a thing a {marked $k{-}1$-hemisphere}. |
1109 Call such a thing a {marked $k{-}1$-hemisphere}. |
1114 |
1110 |
1115 \begin{lem} |
1111 \begin{lem} |
1116 \label{lem:hemispheres} |
1112 \label{lem:hemispheres} |
1117 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
1113 {For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
1118 the category of marked $k$-hemispheres and |
1114 the category of marked $k$-hemispheres and |
1119 homeomorphisms to the category of sets and bijections.} |
1115 homeomorphisms to the category of sets and bijections.} |
1120 \end{lem} |
1116 \end{lem} |
1121 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1117 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1122 We use the same type of colimit construction. |
1118 We use the same type of colimit construction. |
1123 |
1119 |
1124 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
1120 In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
1125 |
1121 |
1126 \begin{module-axiom}[Module boundaries (maps)] |
1122 \begin{module-axiom}[Module boundaries (maps)] |
1127 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
1123 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
1128 These maps, for various $M$, comprise a natural transformation of functors.} |
1124 These maps, for various $M$, comprise a natural transformation of functors.} |
1129 \end{module-axiom} |
1125 \end{module-axiom} |
1130 |
1126 |
1131 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1127 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1132 |
1128 |
1133 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1129 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1134 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
1130 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
1135 and $c\in \cC(\bd M)$. |
1131 and $c\in \cC(\bd M)$. |
1136 |
1132 |
1137 \begin{lem}[Boundary from domain and range] |
1133 \begin{lem}[Boundary from domain and range] |
1138 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1134 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1139 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1135 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1140 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1136 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1141 two maps $\bd: \cM(M_i)\to \cM(E)$. |
1137 two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
1142 Then we have an injective map |
1138 Then we have an injective map |
1143 \[ |
1139 \[ |
1144 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
1140 \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
1145 \] |
1141 \] |
1146 which is natural with respect to the actions of homeomorphisms.} |
1142 which is natural with respect to the actions of homeomorphisms.} |
1147 \end{lem} |
1143 \end{lem} |
1148 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1144 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1149 |
1145 |
1150 Let $\cM(H)_E$ denote the image of $\gl_E$. |
1146 Let $\cl\cM(H)_E$ denote the image of $\gl_E$. |
1151 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
1147 We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
1152 |
1148 |
1153 |
1149 \begin{lem}[Module to category restrictions] |
1154 \begin{module-axiom}[Module to category restrictions] |
|
1155 {For each marked $k$-hemisphere $H$ there is a restriction map |
1150 {For each marked $k$-hemisphere $H$ there is a restriction map |
1156 $\cM(H)\to \cC(H)$. |
1151 $\cl\cM(H)\to \cC(H)$. |
1157 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1152 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1158 These maps comprise a natural transformation of functors.} |
1153 These maps comprise a natural transformation of functors.} |
1159 \end{module-axiom} |
1154 \end{lem} |
1160 |
1155 |
1161 Note that combining the various boundary and restriction maps above |
1156 Note that combining the various boundary and restriction maps above |
1162 (for both modules and $n$-categories) |
1157 (for both modules and $n$-categories) |
1163 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1158 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1164 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
1159 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
1273 \nn{should give figure} |
1268 \nn{should give figure} |
1274 |
1269 |
1275 \begin{module-axiom}[Product (identity) morphisms] |
1270 \begin{module-axiom}[Product (identity) morphisms] |
1276 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
1271 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
1277 $k{+}m$-ball ($m\ge 1$), |
1272 $k{+}m$-ball ($m\ge 1$), |
1278 there is a map $\pi^*:\cC(M)\to \cC(E)$. |
1273 there is a map $\pi^*:\cM(M)\to \cM(E)$. |
1279 These maps must satisfy the following conditions. |
1274 These maps must satisfy the following conditions. |
1280 \begin{enumerate} |
1275 \begin{enumerate} |
1281 \item |
1276 \item |
1282 If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and |
1277 If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and |
1283 if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
1278 if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
1323 \] |
1318 \] |
1324 ($Y$ could be either a marked or plain ball.) |
1319 ($Y$ could be either a marked or plain ball.) |
1325 \end{enumerate} |
1320 \end{enumerate} |
1326 \end{module-axiom} |
1321 \end{module-axiom} |
1327 |
1322 |
|
1323 As in the $n$-category definition, once we have product morphisms we can define |
|
1324 collar maps $\cM(M)\to \cM(M)$. |
|
1325 Note that there are two cases: |
|
1326 the collar could intersect the marking of the marked ball $M$, in which case |
|
1327 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
|
1328 in which case we use a product on a morphism of $\cC$. |
|
1329 |
|
1330 In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
|
1331 $a$ along a map associated to $\pi$. |
|
1332 |
|
1333 \medskip |
1328 |
1334 |
1329 There are two alternatives for the next axiom, according whether we are defining |
1335 There are two alternatives for the next axiom, according whether we are defining |
1330 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
1336 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
1331 In the plain case we require |
1337 In the plain case we require |
1332 |
1338 |
1333 \begin{module-axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$] |
1339 \begin{module-axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$] |
1334 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1340 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1335 to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
1341 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1336 Then $f$ acts trivially on $\cM(M)$.} |
1342 Then $f$ acts trivially on $\cM(M)$.} |
|
1343 In addition, collar maps act trivially on $\cM(M)$. |
1337 \end{module-axiom} |
1344 \end{module-axiom} |
1338 |
|
1339 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
1340 |
1345 |
1341 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
1346 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
1342 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
1347 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
1343 on $\bd B \setmin N$. |
1348 on $\bd B \setmin N$. |
1344 |
1349 |
1345 For $A_\infty$ modules we require |
1350 For $A_\infty$ modules we require |
1346 |
1351 |
1347 \addtocounter{module-axiom}{-1} |
1352 \addtocounter{module-axiom}{-1} |
1348 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
1353 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
1349 {For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1354 For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1350 \[ |
1355 \[ |
1351 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1356 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1352 \] |
1357 \] |
1353 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
1358 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
1354 which fix $\bd M$. |
1359 which fix $\bd M$. |
1355 These action maps are required to be associative up to homotopy |
1360 These action maps are required to be associative up to homotopy, |
1356 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
1361 and also compatible with composition (gluing) in the sense that |
1357 a diagram like the one in Proposition \ref{CHprop} commutes. |
1362 a diagram like the one in Proposition \ref{CHprop} commutes. |
1358 \nn{repeat diagram here?} |
|
1359 \nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
1360 \end{module-axiom} |
1363 \end{module-axiom} |
|
1364 |
|
1365 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1361 |
1366 |
1362 \medskip |
1367 \medskip |
1363 |
1368 |
1364 Note that the above axioms imply that an $n$-category module has the structure |
1369 Note that the above axioms imply that an $n$-category module has the structure |
1365 of an $n{-}1$-category. |
1370 of an $n{-}1$-category. |
1366 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1371 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1367 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1372 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1368 above the non-marked boundary component of $J$. |
1373 above the non-marked boundary component of $J$. |
1369 (More specifically, we collapse $X\times P$ to a single point, where |
1374 (More specifically, we collapse $X\times P$ to a single point, where |
1370 $P$ is the non-marked boundary component of $J$.) |
1375 $P$ is the non-marked boundary component of $J$.) |
1371 \nn{give figure for this?} |
|
1372 Then $\cE$ has the structure of an $n{-}1$-category. |
1376 Then $\cE$ has the structure of an $n{-}1$-category. |
1373 |
1377 |
1374 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1378 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1375 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
1379 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
1376 In this case ($k=1$ and oriented or Spin), there are two types |
1380 In this case ($k=1$ and oriented or Spin), there are two types |