diff -r ada83e7228eb -r 7b0a43bdd3c4 blob1.tex --- a/blob1.tex Tue Jul 01 01:53:15 2008 +0000 +++ b/blob1.tex Tue Jul 01 04:00:22 2008 +0000 @@ -229,7 +229,7 @@ \nn{maybe should look for better name; but this is the name I use elsewhere} is a collection of functors $\cC$ from manifolds of dimension $n$ or less to sets. -These functors must satisfy various properties (see KW TQFT notes for details). +These functors must satisfy various properties (see \cite{kw:tqft} for details). For example: there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; there is a restriction map $\cC(X) \to \cC(\bd X)$; @@ -381,7 +381,7 @@ \nn{Roughly, these are (1) the local relations imply (extended) isotopy; (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). -See KW TQFT notes for details. Need to transfer details to here.} +See \cite{kw:tqft} for details. Need to transfer details to here.} For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, where $a$ and $b$ are maps (fields) which are homotopic rel boundary. @@ -403,7 +403,7 @@ the $n$-manifold $Y$ modulo local relations. The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations is defined to be the dual of $A(Y; c)$. -(See KW TQFT notes or xxxx for details.) +(See \cite{kw:tqft} or xxxx for details.) The blob complex is in some sense the derived version of $A(Y; c)$. @@ -918,15 +918,111 @@ \subsection{`Topological' $A_\infty$ $n$-categories} \label{sec:topological-A-infty}% -This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$ -$n$-category}. The main result of this section is +This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. +The main result of this section is \begin{thm} -Topological $A_\infty$ $1$-categories are equivalent to `standard' -$A_\infty$ $1$-categories. +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. 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?} +\begin{defn} +\label{defn:topological-algebra}% +A ``topological $A_\infty$-algebra'' $A$ consists of the data +\begin{enumerate} +\item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$, +\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, +\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, +\item and whenever $\bdy J \cap \bdy J'$ is a single point, a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, +\end{enumerate} +satisfying the following conditions. +\begin{itemize} +\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 +\begin{equation*} +\xymatrix{ +\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} & +\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ +\CD{J} \tensor A(J) \ar[r]_{\ev_J} & +A(J) +} +\end{equation*} +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 $\Diff(J)$.) +\item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram +\begin{equation*} +\xymatrix{ +A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} && +A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ +A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && +A(J \cup J' \cup J'') +} +\end{equation*} +commutes. +\end{itemize} +\end{defn} + +\begin{rem} +Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of +intervals and diffeomorphisms between them to the category of complexes of vector spaces. +Further, one can combine the second and third pieces of data, asking instead for a map +\begin{equation*} +\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). +\end{equation*} +(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 +diffeomorphisms in $\CD{J'}$.) +\end{rem} + +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 +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: +\begin{equation*} +\gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). +\end{equation*} +The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions. + +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 data +\begin{enumerate} +\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, +\item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ +\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 +$\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ +\end{enumerate} +satisfying the obvious analogous conditions as in Definition \ref{defn:topological-algebra}. +\end{defn} + +\todo{Bimodules, and gluing} + +\todo{the motivating example $C_*(\maps(X, M))$} + +\todo{higher $n$} + + +\newcommand{\skel}[1]{\operatorname{skeleton}(#1)} + +Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your +favorite diffeomorphism $\phi: I \cup I \to I$. +\begin{defn} +We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by +\begin{equation*} +m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I). +\end{equation*} +Next, we define all the `higher associators' $m_k$ by +\todo{} +\end{defn} + +Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should +think of this at the generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition +in the case the $A$ is actually an associative category. +\begin{defn} +\end{defn} \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty @@ -1196,8 +1292,8 @@ \input{text/explicit.tex} % ---------------------------------------------------------------- -\newcommand{\urlprefix}{} -\bibliographystyle{gtart} +%\newcommand{\urlprefix}{} +\bibliographystyle{plain} %Included for winedt: %input "bibliography/bibliography.bib" \bibliography{bibliography/bibliography}