--- a/text/intro.tex Fri Oct 30 21:07:40 2009 +0000
+++ b/text/intro.tex Sat Oct 31 00:20:10 2009 +0000
@@ -6,11 +6,18 @@
\begin{itemize}
\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See \S \ref{sec:fields} \nn{more specific}.)
\item When $n=1$, $\cC$ is just an associative algebroid, and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.)
-\item When $\cC = k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have $$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$
+\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have
+that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains
+on the configurations space of unlabeled points in $M$.
+%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$
\end{itemize}
-The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CD{M}$, extending the usual $\Diff(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a `gluing formula' allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
+The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CD{M}$,
+\nn{maybe replace Diff with Homeo?}
+extending the usual $\Diff(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
-The blob complex definition is motivated by the desire for a `derived' analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of 'resolution'), and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}.
+The blob complex definition is motivated by the desire for a `derived' analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of `resolution'),
+\nn{are the quotes around `derived' and `resolution' necessary?}
+and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}.
We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail.
@@ -22,12 +29,16 @@
Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It's not clear that we could remove the duality conditions from our definition, even if we wanted to.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
\nn{Not sure that the next para is appropriate here}
-The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible; a topological $n$-category associates a vector space to every $B$ diffeomorphic to the $n$-ball. This vector spaces glue together associatively. For an $A_\infty$ $n$-category, we instead associate a chain complex to each such $B$. We require that diffeomorphisms (or the complex of singular chains of diffeomorphisms in the $A_\infty$ case) act. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls, using a topological $n$-category, and the complex $\CM{-}{X}$ of maps to a fix target space $X$.
+The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible; a topological $n$-category associates a vector space to every $B$ diffeomorphic to the $n$-ball. This vector spaces glue together associatively. For an $A_\infty$ $n$-category, we instead associate a chain complex to each such $B$. We require that diffeomorphisms (or the complex of singular chains of diffeomorphisms in the $A_\infty$ case) act. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls, using a topological $n$-category, and the complex $\CM{-}{X}$ of maps to a fixed target space $X$.
+\nn{we might want to make our choice of notation here ($B$, $X$) consistent with later sections ($X$, $T$), or vice versa}
In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $n$-manifold and an $A_\infty$ $n$-category. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
+\nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.}
+
Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CD{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
+
\nn{some more things to cover in the intro}
\begin{itemize}
\item related: we are being unsophisticated from a homotopy theory point of
@@ -137,7 +148,8 @@
\begin{property}[Contractibility]
\label{property:contractibility}%
-\todo{Err, requires a splitting?}
+\nn{this holds with field coefficients, or more generally when
+the map to 0-th homology has a splitting; need to fix statement}
The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$.
\begin{equation}
\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))}
@@ -201,7 +213,9 @@
\begin{property}[Blob complexes of balls form an $A_\infty$ $n$-category]
\label{property:blobs-ainfty}
- Let $\cC$ be a topological $n$-category. Let $Y$ be a $n-k$-manifold. Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m \leq k$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category
+Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold.
+Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m \leq k$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category.
+\nn{the subscript * is only appropriate when $m=k$. }
\end{property}
\begin{rem}
Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category.
@@ -264,7 +278,10 @@
\subsection{Future directions}
\label{sec:future}
-Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
+Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard).
+In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do.
+\nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories}
+More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details.
@@ -273,3 +290,13 @@
\subsection{Thanks and acknowledgements}
We'd like to thank David Ben-Zvi, Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{probably lots more} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.
+
+
+\medskip\hrule\medskip
+
+Still to do:
+\begin{itemize}
+\item say something about starting with semisimple n-cat (trivial?? not trivial?)
+\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations.
+\end{itemize}
+