# HG changeset patch # User Kevin Walker # Date 1280199463 14400 # Node ID 7caafccef7e8d19d27865917a89f14a889403020 # Parent a26808b5db6647b917b5f2c618755d26afb15cf5 starting to revise intro diff -r a26808b5db66 -r 7caafccef7e8 blob1.tex --- a/blob1.tex Fri Jul 23 20:13:19 2010 -0600 +++ b/blob1.tex Mon Jul 26 22:57:43 2010 -0400 @@ -16,7 +16,7 @@ \maketitle -[revision $\ge$ 456; $\ge$ 19 July 2010] +[revision $\ge$ 481; $\ge$ 26 July 2010] {\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}} We're in the midst of revising this, and hope to have a version on the arXiv soon. diff -r a26808b5db66 -r 7caafccef7e8 text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Fri Jul 23 20:13:19 2010 -0600 +++ b/text/appendixes/comparing_defs.tex Mon Jul 26 22:57:43 2010 -0400 @@ -105,7 +105,7 @@ the same thing as traditional modules for traditional 1-categories. -\subsection{Plain 2-categories} +\subsection{Pivotal 2-categories} \label{ssec:2-cats} Let $\cC$ be a topological 2-category. We will construct from $\cC$ a traditional pivotal 2-category. diff -r a26808b5db66 -r 7caafccef7e8 text/intro.tex --- a/text/intro.tex Fri Jul 23 20:13:19 2010 -0600 +++ b/text/intro.tex Mon Jul 26 22:57:43 2010 -0400 @@ -2,10 +2,12 @@ \section{Introduction} -We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. -This blob complex provides a simultaneous generalization of several well-understood constructions: +We construct a chain complex $\bc_*(M; \cC)$ --- the ``blob complex'' --- +associated to an $n$-manifold $M$ and a linear $n$-category with strong duality $\cC$. +This blob complex provides a simultaneous generalization of several well known constructions: \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 $\cC$. +\item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual +topological quantum field theory invariant of $M$ associated to $\cC$. (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. @@ -13,40 +15,48 @@ \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) -%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ +\item When $\cC$ is $\pi^\infty_{\leq n}(T)$, the $A_\infty$ version of the fundamental $n$-groupoid of +the space $T$ (Example \ref{ex:chains-of-maps-to-a-space}), +$\bc_*(M; \cC)$ is homotopy equivalent to $C_*(\Maps(M\to T))$, +the singular chains on the space of maps from $M$ to $T$. +(See Theorem \ref{thm:map-recon}.) \end{itemize} + 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), +(replacing the 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 gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. +One can think of it as the push-out of these two familiar constructions. +More detailed motivations are described in \S \ref{sec:motivations}. The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, -extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing -formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}). +extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (Theorem \ref{thm:evaluation}) and a gluing +formula allowing calculations by cutting manifolds into smaller parts (Theorem \ref{thm:gluing}). -We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. +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. + \subsection{Structure of the paper} The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), -summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) +summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) and outline anticipated future directions (see \S \ref{sec:future}). +\nn{recheck this list after done editing intro} The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. -There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is +There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. -We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. +We sketch the construction of a system of fields from a *-$1$-category and from a pivotal $2$-category. 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 (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. -(It appears that removing the duality conditions from our definition would make it more complicated rather than less.) +(Removing the duality conditions from our definition would make it more complicated rather than less.) 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. @@ -59,7 +69,11 @@ topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category -(using a colimit along certain decompositions of a manifold into balls). With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ with the system of fields constructed from the $n$-category $\cC$. In \S \ref{sec:ainfblob} we give an alternative definition +(using a colimit along certain decompositions of a manifold into balls). +With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ +with the system of fields constructed from the $n$-category $\cC$. +\nn{KW: I don't think we use this notational convention any more, right?} +In \S \ref{sec:ainfblob} we give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). 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. @@ -112,23 +126,26 @@ Finally, later sections address other topics. Section \S \ref{sec:deligne} gives -a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. +a higher dimensional generalization of the Deligne conjecture +(that the little discs operad acts on Hochschild cochains) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", and make connections between our definitions of $n$-categories and familiar 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. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, +as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. +Appendix \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 $M$. +Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) +\nn{...} + -\nn{some more things to cover in the intro} -\begin{itemize} -\item related: we are being unsophisticated from a homotopy theory point of -view and using chain complexes in many places where we could get by with spaces -\item ? one of the points we make (far) below is that there is not really much -difference between (a) systems of fields and local relations and (b) $n$-cats; -thus we tend to switch between talking in terms of one or the other -\end{itemize} +%\item related: we are being unsophisticated from a homotopy theory point of +%view and using chain complexes in many places where we could get by with spaces -\medskip\hrule\medskip +%\item ? one of the points we make (far) below is that there is not really much +%difference between (a) systems of fields and local relations and (b) $n$-cats; +%thus we tend to switch between talking in terms of one or the other + + \subsection{Motivations} \label{sec:motivations}