685 Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
687 Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
686 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
688 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
687 an n-cat} |
689 an n-cat} |
688 } |
690 } |
689 |
691 |
690 \begin{example}[Maps to a space, with a fiber] |
692 \begin{example}[Maps to a space, with a fiber] \label{ex:maps-with-fiber} |
691 \rm |
693 \rm |
692 \label{ex:maps-to-a-space-with-a-fiber}% |
694 \label{ex:maps-to-a-space-with-a-fiber}% |
693 We can modify the example above, by fixing a |
695 We can modify the example above, by fixing a |
694 closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
696 closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
695 otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
697 otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
875 --- composition and $\Diff(X\to X')$ action --- |
877 --- composition and $\Diff(X\to X')$ action --- |
876 also comes from the $\cE\cB_n$ action on $A$. |
878 also comes from the $\cE\cB_n$ action on $A$. |
877 \nn{should we spell this out?} |
879 \nn{should we spell this out?} |
878 |
880 |
879 \nn{Should remark that this is just Lurie's topological chiral homology construction |
881 \nn{Should remark that this is just Lurie's topological chiral homology construction |
880 applied to $n$-balls (check this). |
882 applied to $n$-balls (need to check that colims agree).} |
881 Hmmm... Does Lurie do both framed and unframed cases?} |
|
882 |
883 |
883 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
884 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
884 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
885 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
885 an $\cE\cB_n$-algebra. |
886 an $\cE\cB_n$-algebra. |
886 \nn{The paper is already long; is it worth giving details here?} |
887 \nn{The paper is already long; is it worth giving details here?} |
1061 \begin{proof} |
1062 \begin{proof} |
1062 \nn{...} |
1063 \nn{...} |
1063 \end{proof} |
1064 \end{proof} |
1064 |
1065 |
1065 \nn{need to finish explaining why we have a system of fields; |
1066 \nn{need to finish explaining why we have a system of fields; |
1066 need to say more about ``homological" fields? |
|
1067 (actions of homeomorphisms); |
|
1068 define $k$-cat $\cC(\cdot\times W)$} |
1067 define $k$-cat $\cC(\cdot\times W)$} |
1069 |
1068 |
1070 \subsection{Modules} |
1069 \subsection{Modules} |
1071 |
1070 |
1072 Next we define plain and $A_\infty$ $n$-category modules. |
1071 Next we define plain and $A_\infty$ $n$-category modules. |
1073 The definition will be very similar to that of $n$-categories, |
1072 The definition will be very similar to that of $n$-categories, |
1074 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1073 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1075 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
|
1076 \nn{in particular, need to to get rid of the ``hemisphere axiom"} |
|
1077 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
|
1078 |
1074 |
1079 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1075 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1080 in the context of an $m{+}1$-dimensional TQFT. |
1076 in the context of an $m{+}1$-dimensional TQFT. |
1081 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1077 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1082 This will be explained in more detail as we present the axioms. |
1078 This will be explained in more detail as we present the axioms. |
1083 |
|
1084 \nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.} |
|
1085 |
1079 |
1086 Throughout, we fix an $n$-category $\cC$. |
1080 Throughout, we fix an $n$-category $\cC$. |
1087 For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
1081 For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
1088 We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
1082 We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
1089 |
1083 |
1099 homeomorphisms to the category of sets and bijections.} |
1093 homeomorphisms to the category of sets and bijections.} |
1100 \end{module-axiom} |
1094 \end{module-axiom} |
1101 |
1095 |
1102 (As with $n$-categories, we will usually omit the subscript $k$.) |
1096 (As with $n$-categories, we will usually omit the subscript $k$.) |
1103 |
1097 |
1104 For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
1098 For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1105 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
1099 of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$. |
1106 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1100 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1107 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
1101 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
1108 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
1102 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$ |
|
1103 (see Example \ref{ex:maps-with-fiber}). |
1109 (The union is along $N\times \bd W$.) |
1104 (The union is along $N\times \bd W$.) |
1110 (If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1105 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1111 the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1106 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1112 |
1107 |
1113 \begin{figure}[!ht] |
1108 \begin{figure}[!ht] |
1114 $$\mathfig{.8}{ncat/boundary-collar}$$ |
1109 $$\mathfig{.8}{ncat/boundary-collar}$$ |
1115 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1110 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1116 |
1111 |
1138 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1133 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1139 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
1134 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
1140 and $c\in \cC(\bd M)$. |
1135 and $c\in \cC(\bd M)$. |
1141 |
1136 |
1142 \begin{lem}[Boundary from domain and range] |
1137 \begin{lem}[Boundary from domain and range] |
1143 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
1138 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1144 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
1139 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1145 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1140 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1146 two maps $\bd: \cM(M_i)\to \cM(E)$. |
1141 two maps $\bd: \cM(M_i)\to \cM(E)$. |
1147 Then (axiom) we have an injective map |
1142 Then we have an injective map |
1148 \[ |
1143 \[ |
1149 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
1144 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
1150 \] |
1145 \] |
1151 which is natural with respect to the actions of homeomorphisms.} |
1146 which is natural with respect to the actions of homeomorphisms.} |
1152 \end{lem} |
1147 \end{lem} |
1226 If $k < n$ we require that $\gl_Y$ is injective. |
1221 If $k < n$ we require that $\gl_Y$ is injective. |
1227 (For $k=n$, see below.)} |
1222 (For $k=n$, see below.)} |
1228 \end{module-axiom} |
1223 \end{module-axiom} |
1229 |
1224 |
1230 \begin{module-axiom}[Strict associativity] |
1225 \begin{module-axiom}[Strict associativity] |
1231 {The composition and action maps above are strictly associative.} |
1226 The composition and action maps above are strictly associative. |
1232 \end{module-axiom} |
1227 \end{module-axiom} |
|
1228 |
|
1229 \nn{should say that this is multifold, not just 3-fold} |
1233 |
1230 |
1234 Note that the above associativity axiom applies to mixtures of module composition, |
1231 Note that the above associativity axiom applies to mixtures of module composition, |
1235 action maps and $n$-category composition. |
1232 action maps and $n$-category composition. |
1236 See Figure \ref{zzz1b}. |
1233 See Figure \ref{zzz1b}. |
1237 |
1234 |
1262 \] |
1259 \] |
1263 to $\cM(M)$, |
1260 to $\cM(M)$, |
1264 and these various multifold composition maps satisfy an |
1261 and these various multifold composition maps satisfy an |
1265 operad-type strict associativity condition.} |
1262 operad-type strict associativity condition.} |
1266 |
1263 |
1267 (The above operad-like structure is analogous to the swiss cheese operad |
1264 The above operad-like structure is analogous to the swiss cheese operad |
1268 \cite{MR1718089}.) |
1265 \cite{MR1718089}. |
1269 %\nn{need to double-check that this is true.} |
1266 |
1270 |
1267 \medskip |
1271 \begin{module-axiom}[Product/identity morphisms] |
1268 |
1272 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
1269 We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the |
1273 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
1270 plain ball case. |
1274 If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
1271 Note that a marked pinched product can be decomposed into either |
|
1272 two marked pinched products or a plain pinched product and a marked pinched product. |
|
1273 \nn{should give figure} |
|
1274 |
|
1275 \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 |
|
1277 $k{+}m$-ball ($m\ge 1$), |
|
1278 there is a map $\pi^*:\cC(M)\to \cC(E)$. |
|
1279 These maps must satisfy the following conditions. |
|
1280 \begin{enumerate} |
|
1281 \item |
|
1282 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 |
1275 \[ \xymatrix{ |
1284 \[ \xymatrix{ |
1276 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
1285 E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
1277 M \ar[r]^{f} & M' |
1286 M \ar[r]^{f} & M' |
1278 } \] |
1287 } \] |
1279 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
1288 commutes, then we have |
|
1289 \[ |
|
1290 \pi'^*\circ f = \tilde{f}\circ \pi^*. |
|
1291 \] |
|
1292 \item |
|
1293 Product morphisms are compatible with module composition and module action. |
|
1294 Let $\pi:E\to M$, $\pi_1:E_1\to M_1$, and $\pi_2:E_2\to M_2$ |
|
1295 be pinched products with $E = E_1\cup E_2$. |
|
1296 Let $a\in \cM(M)$, and let $a_i$ denote the restriction of $a$ to $M_i\sub M$. |
|
1297 Then |
|
1298 \[ |
|
1299 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
|
1300 \] |
|
1301 Similarly, if $\rho:D\to X$ is a pinched product of plain balls and |
|
1302 $E = D\cup E_1$, then |
|
1303 \[ |
|
1304 \pi^*(a) = \rho^*(a')\bullet \pi_1^*(a_1), |
|
1305 \] |
|
1306 where $a'$ is the restriction of $a$ to $D$. |
|
1307 \item |
|
1308 Product morphisms are associative. |
|
1309 If $\pi:E\to M$ and $\rho:D\to E$ are marked pinched products then |
|
1310 \[ |
|
1311 \rho^*\circ\pi^* = (\pi\circ\rho)^* . |
|
1312 \] |
|
1313 \item |
|
1314 Product morphisms are compatible with restriction. |
|
1315 If we have a commutative diagram |
|
1316 \[ \xymatrix{ |
|
1317 D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
1318 Y \ar@{^(->}[r] & M |
|
1319 } \] |
|
1320 such that $\rho$ and $\pi$ are pinched products, then |
|
1321 \[ |
|
1322 \res_D\circ\pi^* = \rho^*\circ\res_Y . |
|
1323 \] |
|
1324 ($Y$ could be either a marked or plain ball.) |
|
1325 \end{enumerate} |
1280 \end{module-axiom} |
1326 \end{module-axiom} |
1281 |
1327 |
1282 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
|
1283 |
|
1284 \nn{postpone finalizing the above axiom until the n-cat version is finalized} |
|
1285 |
1328 |
1286 There are two alternatives for the next axiom, according whether we are defining |
1329 There are two alternatives for the next axiom, according whether we are defining |
1287 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
1330 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
1288 In the plain case we require |
1331 In the plain case we require |
1289 |
1332 |