937 |
939 |
938 \medskip |
940 \medskip |
939 |
941 |
940 \nn{say something about associativity here} |
942 \nn{say something about associativity here} |
941 |
943 |
942 \section{Gluing} |
944 \input{text/A-infty.tex} |
943 \label{sec:gluing}% |
945 |
944 |
946 \input{text/gluing.tex} |
945 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
|
946 \begin{itemize} |
|
947 %\mbox{}% <-- gets the indenting right |
|
948 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
|
949 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
950 |
|
951 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
|
952 $A_\infty$ module for $\bc_*(Y \times I)$. |
|
953 |
|
954 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
|
955 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
|
956 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
|
957 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
|
958 \begin{equation*} |
|
959 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
|
960 \end{equation*} |
|
961 \end{itemize} |
|
962 |
|
963 Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
|
964 definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
|
965 and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
|
966 the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras |
|
967 as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the |
|
968 Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the |
|
969 self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs |
|
970 of the gluing formula in the topological context. |
|
971 |
|
972 \subsection{`Topological' $A_\infty$ $n$-categories} |
|
973 \label{sec:topological-A-infty}% |
|
974 |
|
975 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
|
976 The main result of this section is |
|
977 |
|
978 \begin{thm} |
|
979 Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
|
980 $A_\infty$-$1$-categories. |
|
981 \end{thm} |
|
982 |
|
983 Before proving this theorem, we embark upon a long string of definitions. |
|
984 For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define |
|
985 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
|
986 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
|
987 \nn{Something about duals?} |
|
988 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
|
989 \kevin{probably we should say something about the relation |
|
990 to [framed] $E_\infty$ algebras |
|
991 } |
|
992 |
|
993 \todo{} |
|
994 Various citations we might want to make: |
|
995 \begin{itemize} |
|
996 \item \cite{MR2061854} McClure and Smith's review article |
|
997 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
|
998 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
|
999 \item \cite{MR1256989} definition of framed little-discs operad |
|
1000 \end{itemize} |
|
1001 |
|
1002 \begin{defn} |
|
1003 \label{defn:topological-algebra}% |
|
1004 A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
|
1005 \begin{enumerate} |
|
1006 \item For each $1$-manifold $J$ diffeomorphic to the standard interval |
|
1007 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
|
1008 % either roll functoriality into the evaluation map |
|
1009 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
|
1010 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
|
1011 \item For each decomposition of intervals $J = J'\cup J''$, |
|
1012 a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. |
|
1013 % or do it as two separate pieces of data |
|
1014 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
|
1015 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
|
1016 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
|
1017 \end{enumerate} |
|
1018 This data is required to satisfy the following conditions. |
|
1019 \begin{itemize} |
|
1020 \item The evaluation chain map is associative, in that the diagram |
|
1021 \begin{equation*} |
|
1022 \xymatrix{ |
|
1023 & \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\ |
|
1024 \CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\ |
|
1025 & A(J'') & |
|
1026 } |
|
1027 \end{equation*} |
|
1028 commutes up to homotopy. |
|
1029 Here the map $$\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$$ is a composition: take products of singular chains first, then compose diffeomorphisms. |
|
1030 %% or the version for separate pieces of data: |
|
1031 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
|
1032 %\item The evaluation chain map is associative, in that the diagram |
|
1033 %\begin{equation*} |
|
1034 %\xymatrix{ |
|
1035 %\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & |
|
1036 %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
|
1037 %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
|
1038 %A(J) |
|
1039 %} |
|
1040 %\end{equation*} |
|
1041 %commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.) |
|
1042 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
|
1043 \begin{equation*} |
|
1044 \xymatrix{ |
|
1045 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} && |
|
1046 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
|
1047 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
|
1048 A(J \cup J' \cup J'') |
|
1049 } |
|
1050 \end{equation*} |
|
1051 commutes. |
|
1052 \item The gluing and evaluation maps are compatible. |
|
1053 \nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?} |
|
1054 \end{itemize} |
|
1055 \end{defn} |
|
1056 |
|
1057 \begin{rem} |
|
1058 We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
|
1059 constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
|
1060 Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
|
1061 \end{rem} |
|
1062 |
|
1063 %% if we do things separately, we should say this: |
|
1064 %\begin{rem} |
|
1065 %Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
|
1066 %intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
|
1067 %Further, one can combine the second and third pieces of data, asking instead for a map |
|
1068 %\begin{equation*} |
|
1069 %\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
|
1070 %\end{equation*} |
|
1071 %(Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of |
|
1072 %diffeomorphisms in $\CD{J'}$.) |
|
1073 %\end{rem} |
|
1074 |
|
1075 To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each |
|
1076 interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up: |
|
1077 \begin{equation*} |
|
1078 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
|
1079 \end{equation*} |
|
1080 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
|
1081 \todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} |
|
1082 |
|
1083 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
|
1084 \begin{defn} |
|
1085 Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
|
1086 \begin{enumerate} |
|
1087 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
|
1088 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
|
1089 \begin{align*} |
|
1090 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
|
1091 \end{align*} |
|
1092 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, |
|
1093 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
|
1094 \end{enumerate} |
|
1095 The associativity conditions are trivially satisfied. |
|
1096 \end{defn} |
|
1097 |
|
1098 The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
|
1099 Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties |
|
1100 \ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. |
|
1101 |
|
1102 The definition of a module follows closely the definition of an algebra or category. |
|
1103 \begin{defn} |
|
1104 \label{defn:topological-module}% |
|
1105 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
|
1106 consists of the following data. |
|
1107 \begin{enumerate} |
|
1108 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces. |
|
1109 \item For each pair of such marked intervals, |
|
1110 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
|
1111 \item For each decomposition $K = J\cup K'$ of the marked interval |
|
1112 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
|
1113 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
|
1114 \end{enumerate} |
|
1115 The above data is required to satisfy |
|
1116 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
|
1117 \end{defn} |
|
1118 |
|
1119 For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as |
|
1120 a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. |
|
1121 For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
|
1122 (Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |
|
1123 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
|
1124 |
|
1125 The definition of a bimodule is like the definition of a module, |
|
1126 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
|
1127 on the upper boundary and the other with a marked point on the lower boundary. |
|
1128 There are evaluation maps corresponding to gluing unmarked intervals |
|
1129 to the unmarked ends of $K$ and $L$. |
|
1130 |
|
1131 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
|
1132 codimension-0 submanifold of $\bdy X$. |
|
1133 Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
|
1134 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
|
1135 |
|
1136 Next we define the coend |
|
1137 (or gluing or tensor product or self tensor product, depending on the context) |
|
1138 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. |
|
1139 \begin{defn} |
|
1140 We define a category $\cG(M)$. Objects consist of the following data. |
|
1141 \begin{itemize} |
|
1142 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
|
1143 \item For each pair of intervals $N,N'$ an evaluation chain map |
|
1144 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
|
1145 \item For each decomposition of intervals $N = K\cup L$, |
|
1146 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
|
1147 \end{itemize} |
|
1148 This data must satisfy the following conditions. |
|
1149 \begin{itemize} |
|
1150 \item The evaluation maps are associative. |
|
1151 \nn{up to homotopy?} |
|
1152 \item Gluing is strictly associative. |
|
1153 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
|
1154 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
|
1155 agree. |
|
1156 \item the gluing and evaluation maps are compatible. |
|
1157 \end{itemize} |
|
1158 |
|
1159 A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked, |
|
1160 satisfying the following conditions. |
|
1161 \begin{itemize} |
|
1162 \item For each pair of intervals $N,N'$, the diagram |
|
1163 \begin{equation*} |
|
1164 \xymatrix{ |
|
1165 \CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\ |
|
1166 C(N) \ar[r]_{f_N} & C'(N) |
|
1167 } |
|
1168 \end{equation*} |
|
1169 commutes. |
|
1170 \item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition |
|
1171 $$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ |
|
1172 \end{itemize} |
|
1173 \end{defn} |
|
1174 |
|
1175 We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$, |
|
1176 there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$ |
|
1177 factor through the gluing maps for $\gl(M)$. |
|
1178 |
|
1179 We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself |
|
1180 is essentially $C_*(\Maps(S^1 \to M))$. \todo{} |
|
1181 |
|
1182 For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment |
|
1183 $$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. |
|
1184 Showing that it is an initial object is the content of the gluing theorem proved below. |
|
1185 |
|
1186 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
|
1187 $n=1$ case. |
|
1188 One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
|
1189 Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |
|
1190 |
|
1191 \nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.} |
|
1192 |
|
1193 \todo{the motivating example $C_*(\maps(X, M))$} |
|
1194 |
|
1195 |
|
1196 |
|
1197 \newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
|
1198 |
|
1199 Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |
|
1200 favorite diffeomorphism $\phi: I \cup I \to I$. |
|
1201 \begin{defn} |
|
1202 We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by |
|
1203 \begin{equation*} |
|
1204 m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I). |
|
1205 \end{equation*} |
|
1206 Next, we define all the `higher associators' $m_k$ by |
|
1207 \todo{} |
|
1208 \end{defn} |
|
1209 |
|
1210 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
|
1211 think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
|
1212 in the case the $A$ is actually an associative category. |
|
1213 |
|
1214 We'll first define $\cT_{k,n}$ to be the set of planar forests consisting of $n-k$ trees, with a total of $n$ leaves. Thus |
|
1215 \todo{$\cT_{0,n}$ has 1 element, with $n$ vertical lines, $\cT_{1,n}$ has $n-1$ elements, each with a single trivalent vertex, $\cT_{2,n}$ etc...} |
|
1216 \begin{align*} |
|
1217 \end{align*} |
|
1218 |
|
1219 \begin{defn} |
|
1220 The topological $A_\infty$ category $\bc_*^A$ is doubly graded, by `blob degree' and `internal degree'. We'll write $\bc_k^A$ for the blob degree $k$ piece. |
|
1221 The homological degree of an element $a \in \bc_*^A(J)$ |
|
1222 is the sum of the blob degree and the internal degree. |
|
1223 |
|
1224 We first define $\bc_0^A(J)$ as a vector space by |
|
1225 \begin{equation*} |
|
1226 \bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
|
1227 \end{equation*} |
|
1228 (That is, for each division of $J$ into finitely many subintervals, |
|
1229 we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
|
1230 and a copy of $A$ for each subinterval.) |
|
1231 The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains |
|
1232 plus the sum of the homological degrees of the elements of $A$. |
|
1233 The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$. |
|
1234 |
|
1235 Next, |
|
1236 \begin{equation*} |
|
1237 \bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
|
1238 \end{equation*} |
|
1239 \end{defn} |
|
1240 |
|
1241 \begin{figure}[!ht] |
|
1242 \begin{equation*} |
|
1243 \mathfig{0.7}{associahedron/A4-vertices} |
|
1244 \end{equation*} |
|
1245 \caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.} |
|
1246 \label{fig:A4-vertices} |
|
1247 \end{figure} |
|
1248 |
|
1249 \begin{figure}[!ht] |
|
1250 \begin{equation*} |
|
1251 \mathfig{0.7}{associahedron/A4-faces} |
|
1252 \end{equation*} |
|
1253 \caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.} |
|
1254 \label{fig:A4-vertices} |
|
1255 \end{figure} |
|
1256 |
|
1257 \newcommand{\tm}{\widetilde{m}} |
|
1258 |
|
1259 Let $\tm_1(a) = a$. |
|
1260 |
|
1261 We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it. |
|
1262 \begin{align} |
|
1263 \notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\ |
|
1264 \label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\ |
|
1265 \label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\ |
|
1266 \label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) |
|
1267 \end{align} |
|
1268 The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$. |
|
1269 The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves. |
|
1270 Note here that we have one more leaf than there arguments of $\tm_k$. |
|
1271 (See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.) |
|
1272 We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves. |
|
1273 The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both |
|
1274 vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form |
|
1275 $$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$ |
|
1276 where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices. |
|
1277 If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$ |
|
1278 in \eqref{eq:bdy-tm-k-3}, |
|
1279 where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex. |
|
1280 For example, we have |
|
1281 \begin{align*} |
|
1282 \bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\ |
|
1283 & \qquad - a \tensor b + m_2(a \tensor b) \\ |
|
1284 \bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\ |
|
1285 & \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ |
|
1286 & \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) |
|
1287 \end{align*} |
|
1288 \begin{align*} |
|
1289 \bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\ |
|
1290 & + \left(\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)\right) + \\ |
|
1291 & + \left(\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d))\right. + \\ |
|
1292 & + \left.\tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)\right) \\ |
|
1293 \end{align*} |
|
1294 See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest |
|
1295 to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one. |
|
1296 |
|
1297 \begin{figure}[!ht] |
|
1298 \begin{equation*} |
|
1299 \mathfig{1.0}{associahedron/A4-terms} |
|
1300 \end{equation*} |
|
1301 \caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.} |
|
1302 \label{fig:A4-terms} |
|
1303 \end{figure} |
|
1304 |
|
1305 \begin{lem} |
|
1306 This definition actually results in a chain complex, that is $\bdy^2 = 0$. |
|
1307 \end{lem} |
|
1308 \begin{proof} |
|
1309 \newcommand{\T}{\text{---}} |
|
1310 \newcommand{\ssum}[1]{{\sum}^{(#1)}} |
|
1311 For the duration of this proof, inside a summation over variables $l_1, \ldots, l_m$, an expression with $m$ dashes will be interpreted |
|
1312 by replacing each dash with contiguous factors from $a_1 \tensor \cdots \tensor a_k$, so the first dash takes the first $l_1$ factors, the second |
|
1313 takes the next $l_2$ factors, and so on. Further, we'll write $\ssum{m}$ for $\sum_{\sum_{i=1}^m l_i = k}$. |
|
1314 In this notation, the formula for the differential becomes |
|
1315 \begin{align} |
|
1316 \notag |
|
1317 \bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
|
1318 \intertext{and we calculate} |
|
1319 \notag |
|
1320 \bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
|
1321 \notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
|
1322 \notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
|
1323 \label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\ |
|
1324 \label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\ |
|
1325 \label{eq:d23} & \qquad + \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2+l_3} \sigma_{l_1;l_2,l_3} \\ |
|
1326 \label{eq:d24} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \sigma_{0;l_1,l_2+l_3+l_4} \tau_{l_1;l_2,l_3,l_4} \\ |
|
1327 \label{eq:d25} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \tau_{0;l_1,l_2,l_3+l_4} ??? \\ |
|
1328 \label{eq:d26} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2,l_3,l_4} \sigma_{0;l_1,l_2} \\ |
|
1329 \label{eq:d27} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2+l_3,l_4,l_5} \tau_{0;l_1,l_2,l_3} \\ |
|
1330 \label{eq:d28} & \qquad + \ssum{5} \tm(\T \tensor m(\T \tensor m(\T) \tensor \T) \tensor \T) \times \tau_{0;l_1,l_2+l_3+l_4,l_5} ??? \\ |
|
1331 \label{eq:d29} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3+l_4+l_5} ??? |
|
1332 \end{align} |
|
1333 Now, we see the the expressions on the right hand side of line \eqref{eq:d21} and those on \eqref{eq:d23} cancel. Similarly, line \eqref{eq:d22} cancels |
|
1334 with \eqref{eq:d25}, \eqref{eq:d24} with \eqref{eq:d26}, and \eqref{eq:d27} with \eqref{eq:d29}. Finally, we need to see that \eqref{eq:d28} gives $0$, |
|
1335 by the usual relations between the $m_k$ in an $A_\infty$ algebra. |
|
1336 \end{proof} |
|
1337 |
|
1338 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
|
1339 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
|
1340 easy, I think, so maybe it should be done earlier??} |
|
1341 |
|
1342 \bigskip |
|
1343 |
|
1344 Outline: |
|
1345 \begin{itemize} |
|
1346 \item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
|
1347 use graphical/tree point of view, rather than following Keller exactly |
|
1348 \item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
|
1349 \item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
|
1350 \item motivating example: $C_*(\maps(X, M))$ |
|
1351 \item maybe incorporate dual point of view (for $n=1$), where points get |
|
1352 object labels and intervals get 1-morphism labels |
|
1353 \end{itemize} |
|
1354 |
|
1355 |
|
1356 \subsection{$A_\infty$ action on the boundary} |
|
1357 \label{sec:boundary-action}% |
|
1358 Let $Y$ be an $n{-}1$-manifold. |
|
1359 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
|
1360 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
|
1361 of an $A_\infty$ category. |
|
1362 |
|
1363 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
|
1364 $I\cup I \cong I$. Given this choice, gluing gives a map |
|
1365 \eq{ |
|
1366 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
|
1367 \cong \bc_*(Y\times I; a, c) |
|
1368 } |
|
1369 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
|
1370 higher associators of the $A_\infty$ structure, more or less canonically. |
|
1371 |
|
1372 \nn{is this obvious? does more need to be said?} |
|
1373 |
|
1374 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
|
1375 |
|
1376 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
|
1377 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
|
1378 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
|
1379 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
|
1380 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
|
1381 of $Y$ in $X$. |
|
1382 |
|
1383 In the next section we use the above $A_\infty$ actions to state and prove |
|
1384 a gluing theorem for the blob complexes of $n$-manifolds. |
|
1385 |
|
1386 |
|
1387 \subsection{The gluing formula} |
|
1388 \label{sec:gluing-formula}% |
|
1389 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
|
1390 of $Y \du -Y$ contained in its boundary. |
|
1391 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
|
1392 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
|
1393 of $X$. |
|
1394 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
|
1395 where $c\sgl \in \cC(\bd X\sgl)$, |
|
1396 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
|
1397 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
|
1398 |
|
1399 \begin{thm} |
|
1400 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
|
1401 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
|
1402 \end{thm} |
|
1403 |
|
1404 The proof will occupy the remainder of this section. |
|
1405 |
|
1406 \nn{...} |
|
1407 |
|
1408 \bigskip |
|
1409 |
|
1410 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
|
1411 |
|
1412 |
|
1413 |
947 |
1414 |
948 |
1415 |
949 |
1416 \section{Commutative algebras as $n$-categories} |
950 \section{Commutative algebras as $n$-categories} |
1417 |
951 |