742 |
742 |
743 |
743 |
744 |
744 |
745 \subsection{Modules} |
745 \subsection{Modules} |
746 |
746 |
747 Next we define topological and $A_\infty$ $n$-category modules. |
747 Next we define plain and $A_\infty$ $n$-category modules. |
748 The definition will be very similar to that of $n$-categories, |
748 The definition will be very similar to that of $n$-categories, |
749 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
749 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
750 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
750 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
751 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
751 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
752 |
752 |
753 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
753 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
754 in the context of an $m{+}1$-dimensional TQFT. |
754 in the context of an $m{+}1$-dimensional TQFT. |
755 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
755 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
756 This will be explained in more detail as we present the axioms. |
756 This will be explained in more detail as we present the axioms. |
|
757 |
|
758 \nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.} |
757 |
759 |
758 Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
760 Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
759 |
761 |
760 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
762 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
761 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
763 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1094 \subsection{Morphisms of $A_\infty$ 1-cat modules} |
1096 \subsection{Morphisms of $A_\infty$ 1-cat modules} |
1095 |
1097 |
1096 In order to state and prove our version of the higher dimensional Deligne conjecture |
1098 In order to state and prove our version of the higher dimensional Deligne conjecture |
1097 (Section \ref{sec:deligne}), |
1099 (Section \ref{sec:deligne}), |
1098 we need to define morphisms of $A_\infty$ 1-cat modules and establish |
1100 we need to define morphisms of $A_\infty$ 1-cat modules and establish |
1099 some elementary properties of these. |
1101 some of their elementary properties. |
1100 |
1102 |
1101 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules |
1103 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules |
1102 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction |
1104 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction |
1103 \begin{eqnarray*} |
1105 \begin{eqnarray*} |
1104 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1106 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1114 and modules $\cM_\cC$ and $_\cC\cN$, |
1116 and modules $\cM_\cC$ and $_\cC\cN$, |
1115 \[ |
1117 \[ |
1116 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1118 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1117 \] |
1119 \] |
1118 |
1120 |
1119 In the next few paragraphs define the things appearing in the above equation: |
1121 In the next few paragraphs we define the things appearing in the above equation: |
1120 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1122 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1121 $\hom_\cC$. |
1123 $\hom_\cC$. |
1122 |
1124 |
1123 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules |
1125 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules |
1124 for general $n$. |
1126 for general $n$. |
1177 (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
1179 (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
1178 \] |
1180 \] |
1179 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
1181 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
1180 to the right-marked interval $J\setmin K$. |
1182 to the right-marked interval $J\setmin K$. |
1181 This extends to a functor from all left-marked intervals (not just those contained in $J$). |
1183 This extends to a functor from all left-marked intervals (not just those contained in $J$). |
|
1184 \nn{need to say more here; not obvious how homeomorphisms act} |
1182 It's easy to verify the remaining module axioms. |
1185 It's easy to verify the remaining module axioms. |
1183 |
1186 |
1184 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1187 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1185 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1188 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1186 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1189 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1199 Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
1202 Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
1200 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1203 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1201 omitted. |
1204 omitted. |
1202 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1205 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1203 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1206 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1204 (See Figure xxxx.) |
1207 (See Figure \ref{fig:lmar}.) |
|
1208 \begin{figure}[t]\begin{equation*} |
|
1209 \mathfig{.6}{tempkw/left-marked-antirefinements} |
|
1210 \end{equation*}\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
1205 |
1211 |
1206 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1212 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1207 The underlying vector space is |
1213 The underlying vector space is |
1208 \[ |
1214 \[ |
1209 \prod_l \prod_{\olD} \hom[l]\left( |
1215 \prod_l \prod_{\olD} \hom[l]\left( |
1240 g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
1246 g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
1241 \] |
1247 \] |
1242 constitutes a null homotopy of |
1248 constitutes a null homotopy of |
1243 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
1249 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
1244 should be interpreted as above). |
1250 should be interpreted as above). |
|
1251 |
|
1252 Define a {\it naive morphism} |
|
1253 \nn{should consider other names for this} |
|
1254 of modules to be a collection of {\it chain} maps |
|
1255 \[ |
|
1256 h_K : \cX(K)\to \cY(K) |
|
1257 \] |
|
1258 for each left-marked interval $K$. |
|
1259 These are required to commute with gluing; |
|
1260 for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
|
1261 \[ \xymatrix{ |
|
1262 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
|
1263 \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
|
1264 \ar[d]^{\gl} \\ |
|
1265 \cX(K) \ar[r]^{h_{K}} & \cY(K) |
|
1266 } \] |
|
1267 Given such an $h$ we can construct a non-naive morphism $g$, with $\bd g = 0$, as follows. |
|
1268 Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
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1269 If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
|
1270 \[ |
|
1271 g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
|
1272 \] |
|
1273 Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
|
1274 If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
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1275 If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
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1276 that each $h_K$ is a chain map. |
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1277 |
|
1278 \medskip |
|
1279 |
|
1280 Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
|
1281 \[ |
|
1282 g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
|
1283 \] |
|
1284 \nn{this is fairly straightforward, but the details are messy enough that I'm inclined |
|
1285 to postpone writing it up, in the hopes that I'll think of a better way to organize things.} |
|
1286 |
|
1287 |
|
1288 |
|
1289 |
|
1290 \medskip |
|
1291 |
1245 |
1292 |
1246 \nn{do we need to say anything about composing morphisms of modules?} |
1293 \nn{do we need to say anything about composing morphisms of modules?} |
1247 |
1294 |
1248 \nn{should we define functors between $n$-cats in a similar way?} |
1295 \nn{should we define functors between $n$-cats in a similar way?} |
1249 |
1296 |