1023 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1023 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1024 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
1024 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
1025 %component $\bd_i W$ of $W$. |
1025 %component $\bd_i W$ of $W$. |
1026 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
1026 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
1027 |
1027 |
1028 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
1028 We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above. |
1029 \nn{give ref} |
1029 (If $k = n$ and our $n$-categories are enriched, then |
1030 (If $k = n$ and our $k$-categories are enriched, then |
|
1031 $\cC(W, \cN)$ will have additional structure; see below.) |
1030 $\cC(W, \cN)$ will have additional structure; see below.) |
1032 |
1031 |
1033 Define a permissible decomposition of $W$ to be a decomposition |
1032 Define a permissible decomposition of $W$ to be a decomposition |
1034 \[ |
1033 \[ |
1035 W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
1034 W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
1037 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
1036 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
1038 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
1037 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
1039 with $M_{ib}\cap Y_i$ being the marking. |
1038 with $M_{ib}\cap Y_i$ being the marking. |
1040 (See Figure \ref{mblabel}.) |
1039 (See Figure \ref{mblabel}.) |
1041 \begin{figure}[!ht]\begin{equation*} |
1040 \begin{figure}[!ht]\begin{equation*} |
1042 \mathfig{.6}{ncat/mblabel} |
1041 \mathfig{.4}{ncat/mblabel} |
1043 \end{equation*}\caption{A permissible decomposition of a manifold |
1042 \end{equation*}\caption{A permissible decomposition of a manifold |
1044 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1043 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1045 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1044 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1046 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1045 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1047 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
1046 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
1048 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
1047 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
1049 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1048 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1050 |
1049 |
1051 $\cN$ determines |
1050 The collection of modules $\cN$ determines |
1052 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
1051 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
1053 (possibly with additional structure if $k=n$). |
1052 (possibly with additional structure if $k=n$). |
1054 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
1053 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
1055 \[ |
1054 \[ |
1056 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
1055 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
1057 \] |
1056 \] |
1058 such that the restrictions to the various pieces of shared boundaries amongst the |
1057 such that the restrictions to the various pieces of shared boundaries amongst the |
1059 $X_a$ and $M_{ib}$ all agree. |
1058 $X_a$ and $M_{ib}$ all agree. |
1060 (Think fibered product.) |
1059 (That is, the fibered product over the boundary maps.) |
1061 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1060 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1062 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
1061 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
1063 |
1062 |
1064 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
1063 We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
1065 (Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
1064 (As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
1066 homotopy colimit.) |
1065 homotopy colimit.) |
1067 |
1066 |
1068 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1067 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1069 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
1068 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
1070 $D\times Y_i \sub \bd(D\times W)$. |
1069 $D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
1071 |
1070 has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
1072 It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
|
1073 has the structure of an $n{-}k$-category. |
|
1074 |
1071 |
1075 \medskip |
1072 \medskip |
1076 |
1073 |
1077 |
1074 |
1078 We will use a simple special case of the above |
1075 We will use a simple special case of the above |
1079 construction to define tensor products |
1076 construction to define tensor products |
1080 of modules. |
1077 of modules. |
1081 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1078 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1082 (If $k=1$ and manifolds are oriented, then one should be |
1079 (If $k=1$ and our manifolds are oriented, then one should be |
1083 a left module and the other a right module.) |
1080 a left module and the other a right module.) |
1084 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
1081 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
1085 Define the tensor product of $\cM_1$ and $\cM_2$ to be the |
1082 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
1086 $n{-}1$-category $\cT(J, \cM_1, \cM_2)$, |
1083 $n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially) |
1087 \[ |
|
1088 \cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) . |
|
1089 \] |
|
1090 This of course depends (functorially) |
|
1091 on the choice of 1-ball $J$. |
1084 on the choice of 1-ball $J$. |
1092 |
1085 |
1093 We will define a more general self tensor product (categorified coend) below. |
1086 We will define a more general self tensor product (categorified coend) below. |
1094 |
1087 |
1095 %\nn{what about self tensor products /coends ?} |
1088 %\nn{what about self tensor products /coends ?} |
1103 |
1096 |
1104 \subsection{Morphisms of $A_\infty$ 1-cat modules} |
1097 \subsection{Morphisms of $A_\infty$ 1-cat modules} |
1105 |
1098 |
1106 In order to state and prove our version of the higher dimensional Deligne conjecture |
1099 In order to state and prove our version of the higher dimensional Deligne conjecture |
1107 (Section \ref{sec:deligne}), |
1100 (Section \ref{sec:deligne}), |
1108 we need to define morphisms of $A_\infty$ 1-cat modules and establish |
1101 we need to define morphisms of $A_\infty$ 1-category modules and establish |
1109 some of their elementary properties. |
1102 some of their elementary properties. |
1110 |
1103 |
1111 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules |
1104 To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1112 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction |
|
1113 \begin{eqnarray*} |
1105 \begin{eqnarray*} |
1114 \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)) \\ |
1115 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1107 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1116 {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
1108 {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
1117 \end{eqnarray*} |
1109 \end{eqnarray*} |
1123 and modules $\cM_\cC$ and $_\cC\cN$, |
1115 and modules $\cM_\cC$ and $_\cC\cN$, |
1124 \[ |
1116 \[ |
1125 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1117 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1126 \] |
1118 \] |
1127 |
1119 |
1128 In the next few paragraphs we define the things appearing in the above equation: |
1120 In the next few paragraphs we define the objects appearing in the above equation: |
1129 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1121 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1130 $\hom_\cC$. |
1122 $\hom_\cC$. |
1131 |
1123 |
1132 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules |
1124 |
|
1125 \def\olD{{\overline D}} |
|
1126 \def\cbar{{\bar c}} |
|
1127 In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
1133 for general $n$. |
1128 for general $n$. |
1134 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
1129 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
1135 and their gluings (antirefinements). |
1130 and their gluings (antirefinements). |
1136 (The tensor product will depend (functorially) on the choice of $J$.) |
1131 (This tensor product depends functorially on the choice of $J$.) |
1137 To a subdivision |
1132 To a subdivision $D$ |
1138 \[ |
1133 \[ |
1139 J = I_1\cup \cdots\cup I_p |
1134 J = I_1\cup \cdots\cup I_p |
1140 \] |
1135 \] |
1141 we associate the chain complex |
1136 we associate the chain complex |
1142 \[ |
1137 \[ |
1143 \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
1138 \psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
1144 \] |
1139 \] |
1145 (If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.) |
|
1146 To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
1140 To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
1147 module actions of $\cC$ on $\cM$ and $\cN$. |
1141 module actions of $\cC$ on $\cM$ and $\cN$. |
1148 \def\olD{{\overline D}} |
|
1149 \def\cbar{{\bar c}} |
|
1150 The underlying graded vector space of the homotopy colimit is |
1142 The underlying graded vector space of the homotopy colimit is |
1151 \[ |
1143 \[ |
1152 \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
1144 \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
1153 \] |
1145 \] |
1154 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
1146 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
1155 runs through chains of antirefinements, and $[l]$ denotes a grading shift. |
1147 runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. |
1156 We will denote an element of the summand indexed by $\olD$ by |
1148 We will denote an element of the summand indexed by $\olD$ by |
1157 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
1149 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
1158 The boundary map is given (ignoring signs) by |
1150 The boundary map is given by |
1159 \begin{eqnarray*} |
1151 \begin{align*} |
1160 \bd(\olD\ot m\ot\cbar\ot n) &=& \olD\ot\bd(m\ot\cbar)\ot n + \olD\ot m\ot\cbar\ot \bd n + \\ |
1152 \bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
1161 & & \;\; (\bd_+ \olD)\ot m\ot\cbar\ot n + (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) , |
1153 & \qquad + (-1)^l \olD\ot\bd(m\ot\cbar\ot n) |
1162 \end{eqnarray*} |
1154 \end{align*} |
1163 where $\bd_+ \olD = \sum_{i>0} (D_0, \cdots \widehat{D_i} \cdots , D_l)$ (the part of the simplicial |
1155 where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
1164 boundary which retains $D_0$), $\bd_0 \olD = (D_1, \cdots , D_l)$, |
1156 boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
1165 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
1157 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
1166 |
1158 |
1167 $(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
1159 $(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
1168 \[ |
1160 \[ |
1169 \prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
1161 \prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
1203 \end{eqnarray*} |
1195 \end{eqnarray*} |
1204 |
1196 |
1205 We are almost ready to give the definition of morphisms between arbitrary modules |
1197 We are almost ready to give the definition of morphisms between arbitrary modules |
1206 $\cX_\cC$ and $\cY_\cC$. |
1198 $\cX_\cC$ and $\cY_\cC$. |
1207 Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
1199 Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
1208 To fix this, we define subdivisions are antirefinements of left-marked intervals. |
1200 To fix this, we define subdivisions as antirefinements of left-marked intervals. |
1209 Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
1201 Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
1210 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1202 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1211 omitted. |
1203 omitted. |
1212 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1204 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1213 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1205 gluing subintervals together and/or omitting some of the rightmost subintervals. |