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