blob: comparison blob1.tex

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 \begin{thm}
 Topological $A_\infty$-$1$-categories are equivalent to `standard'
 $A_\infty$-$1$-categories.
 \end{thm}
 Before proving this theorem, we embark upon a long string of definitions.
-\kevin{the \\kevin macro seems to be truncating text of the left side of the page}
 For expository purposes, we begin with the $n=1$ special cases, and define
 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn
 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules.
 \nn{Something about duals?}
 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?}
 \kevin{probably we should say something about the relation
-to [framed] $E_\infty$ algebras}
+to [framed] $E_\infty$ algebras
+}
+\todo{}
+Various citations we might want to make:
+\begin{itemize}
+\item \cite{MR2061854} McClure and Smith's review article
+\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad)
+\item \cite{MR0236922,MR0420609} Boardman and Vogt
+\item \cite{MR1256989} definition of framed little-discs operad
+\end{itemize}
 \begin{defn}
 \label{defn:topological-algebra}%
 A ``topological $A_\infty$-algebra'' $A$ consists of the following data.
 \begin{enumerate}
 \item For each $1$-manifold $J$ diffeomorphic to the standard interval
 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$.
 % either roll functoriality into the evaluation map
 \item For each pair of intervals $J,J'$ an `evaluation' chain map
 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$.
 \item For each decomposition of intervals $J = J'\cup J''$,
 a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$.
 % or do it as two separate pieces of data
 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$,
 \CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\
 \CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} &
 A(J'')
 }
 \end{equation*}
 commutes.
 \kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy}
 (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.)
 %% or the version for separate pieces of data:
 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same.
 %\item The evaluation chain map is associative, in that the diagram
 \ref{property:evaluation} and \ref{property:gluing-map} respectively.
 The definition of a module follows closely the definition of an algebra or category.
 \begin{defn}
 \label{defn:topological-module}%
 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$
 consists of the following data.
 \begin{enumerate}
 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces.
 \item For each pair of such marked intervals,
 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$.
 \item For each decomposition $K = J\cup K'$ of the marked interval
 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map
 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$.
 \end{enumerate}
 The above data is required to satisfy
 conditions analogous to those in Definition \ref{defn:topological-algebra}.
 \end{defn}
 Any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) becomes a topological $A_\infty$ module over
 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$.
 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point
 on the upper boundary and the other with a marked point on the lower boundary.
 There are evaluation maps corresponding to gluing unmarked intervals
 to the unmarked ends of $K$ and $L$.
 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a
 codimension-0 submanifold of $\bdy X$.
 Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the
 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$.
 Next we define the coend
 (or gluing or tensor product or self tensor product, depending on the context)
 $\gl(M)$ of a topological $A_\infty$ bimodule $M$.
 $\gl(M)$ is defined to be the universal thing with the following structure.
 \begin{itemize}
 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N).
 \item For each pair of intervals $N,N'$ an evaluation chain map
 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$.
 \item For each decomposition of intervals $N = K\cup L$,
 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$.
 \item The evaluation maps are associative.
 \nn{up to homotopy?}
 \item Gluing is strictly associative.
 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to
 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$
 agree.
 \item the gluing and evaluation maps are compatible.
 \end{itemize}
 Bu universal we mean that given any other collection of chain complexes, evaluation maps
 and gluing maps, they factor through the universal thing.
 \nn{need to say this in more detail, in particular give the properties of the factoring map}
 Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment
 $N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described
 in the above bullet points.
 Showing that it is the universal such thing is the content of the gluing theorem proved below.
 The definitions for a topological $A_\infty$-$n$-category are very similar to the above
 $n=1$ case.

changeset 37	2f677e283c26
parent 36	f5e553fbd693
child 40	b7bc1a931b73