232 These functors must satisfy various properties (see KW TQFT notes for details).

   232 These functors must satisfy various properties (see \cite{kw:tqft} for details).

   384 See KW TQFT notes for details.  Need to transfer details to here.}

   384 See \cite{kw:tqft} for details.  Need to transfer details to here.}

   406 (See KW TQFT notes or xxxx for details.)

   406 (See \cite{kw:tqft} or xxxx for details.)

   921 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$

   921 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}.

   922 $n$-category}. The main result of this section is

   922 The main result of this section is

   925 Topological $A_\infty$ $1$-categories are equivalent to `standard'

   925 Topological $A_\infty$-$1$-categories are equivalent to `standard'

   926 $A_\infty$ $1$-categories.

   926 $A_\infty$-$1$-categories.

   929 

   929 Before proving this theorem, we embark upon a long string of definitions. For expository purposes, we begin with the $n=1$ special cases, and define

   930 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn

   931 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules.

   932 \nn{Something about duals?}

   933 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?}

   934 

   935 \begin{defn}

   936 \label{defn:topological-algebra}%

   937 A ``topological $A_\infty$-algebra'' $A$ consists of the data

   938 \begin{enumerate}

   939 \item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$,

   940 \item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$,

   941 \item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$,

   942 \item and whenever $\bdy J \cap \bdy J'$ is a single point, a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$,

   943 \end{enumerate}

   944 satisfying the following conditions.

   945 \begin{itemize}

   946 \item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same.

   947 \item The evaluation chain map is associative, in that the diagram

   948 \begin{equation*}

   949 \xymatrix{

   950 \CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} &

   951 \CD{J} \tensor A(J) \ar[d]^{\ev_J} \\

   952 \CD{J} \tensor A(J) \ar[r]_{\ev_J} &

   953 A(J)

   954 }

   955 \end{equation*}

   956 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 $\Diff(J)$.)

   957 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram

   958 \begin{equation*}

   959 \xymatrix{

   960 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} &&

   961 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\

   962 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} &&

   963 A(J \cup J' \cup J'')

   964 }

   965 \end{equation*}

   966 commutes.

   967 \end{itemize}

   968 \end{defn}

   969 

   970 \begin{rem}

   971 Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of

   972 intervals and diffeomorphisms between them to the category of complexes of vector spaces.

   973 Further, one can combine the second and third pieces of data, asking instead for a map

   974 \begin{equation*}

   975 \ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J').

   976 \end{equation*}

   977 (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

   978 diffeomorphisms in $\CD{J'}$.)

   979 \end{rem}

   980 

   981 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

   982 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:

   983 \begin{equation*}

   984 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+).

   985 \end{equation*}

   986 The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions.

   987 

   988 The definition of a module follows closely the definition of an algebra or category.

   989 \begin{defn}

   990 \label{defn:topological-module}%

   991 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data

   992 \begin{enumerate}

   993 \item a functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a boundary, to complexes of vector spaces,

   994 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$

   995 \item whenever $\bdy J \cap K$ is a single point, and isn't the marked point of $K$ \todo{ugh, that's so gross}, a gluing map

   996 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$

   997 \end{enumerate}

   998 satisfying the obvious analogous conditions as in Definition \ref{defn:topological-algebra}.

   999 \end{defn}

  1000 

  1001 \todo{Bimodules, and gluing}

  1002 

  1003 \todo{the motivating example $C_*(\maps(X, M))$}

  1004 

  1005 \todo{higher $n$}

  1006 

  1007 

  1008 \newcommand{\skel}[1]{\operatorname{skeleton}(#1)}

  1009 

  1010 Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your

  1011 favorite diffeomorphism $\phi: I \cup I \to I$.

  1012 \begin{defn}

  1013 We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by

  1014 \begin{equation*}

  1015 m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I).

  1016 \end{equation*}

  1017 Next, we define all the `higher associators' $m_k$ by

  1018 \todo{}

  1019 \end{defn}

  1020 

  1021 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should

  1022 think of this at the generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition

  1023 in the case the $A$ is actually an associative category.

  1024 \begin{defn}

  1025 \end{defn}

  1199 \newcommand{\urlprefix}{}

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