%!TEX root = ../blob1.tex

\section{Introduction}

We construct a chain complex $\bc_*(M; \cC)$ --- the ``blob complex'' --- 

associated to an $n$-manifold $M$ and a linear $n$-category $\cC$ with strong duality.

This blob complex provides a simultaneous generalization of several well known constructions:

\begin{itemize}

\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 Proposition \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 $\HC_*(\cC)$.

(See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.)

%\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}.)

\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 the quotient of fields by local relations with some sort of resolution), 

and for a generalization of Hochschild homology to higher $n$-categories.

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 $H_0$ (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.

%See \S \ref{sec:future} for slightly more detail.

Throughout, we have resisted the temptation to work in the greatest possible generality.

%(Don't worry, it wasn't that hard.)

In most of the places where we say ``set" or ``vector space", any symmetric monoidal category 

with sufficient limits and colimits would do.

We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories).

\subsection{Structure of the paper}

The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), 

%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 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 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.

(Removing the duality conditions from our definition would make it more complicated rather than less.) 

We call these ``disk-like $n$-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.

(See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.)

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 disk-like $n$-category associates a 

vector space to every $B$ homeomorphic to the $n$-ball.

These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.

For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to 

each such $B$ and ask that the action of 

homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms.

The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: 

the blob complexes of $n$-balls labelled by a 

disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.

In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category 

of sphere modules.

When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners.

In \S \ref{ss:ncat_fields}  we explain how to construct a system of fields from a disk-like $n$-category 

(using a colimit along certain decompositions of a manifold into balls). 

With this in hand, we 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 ordinary $n$-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 Theorem \ref{thm:gluing} below.

The relationship between all these ideas is sketched in Figure \ref{fig:outline}.

% NB: the following tikz requires a *more recent* version of PGF than is distributed with MacTex 2010.

% grab the latest build from http://www.texample.net/tikz/builds/

% unzip it in your personal tex tree, and run "mktexlsr ." there

\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt]

\begin{figure}[t]

{\center

\beginpgfgraphicnamed{gadgets-external}%

\begin{tikzpicture}[align=center,line width = 1.5pt]

\newcommand{\xxa}{2}

\newcommand{\xxb}{8}

\newcommand{\yya}{14}

\newcommand{\yyb}{10}

\newcommand{\yyc}{6}

\node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category};

\node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category};

\node[box] at (\xxb,\yya) (A) {$A(M)$ \\ the (dual) TQFT \\ Hilbert space};

\node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations};

\node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex};

\node[box] at (\xxa,\yyc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category};

\node[box] at (\xxb,\yyc) (BCs) {$\underrightarrow{\cC_*}(M)$};

\draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A);

\draw[->] (FU) -- node[above] {blob complex \\ for $M$} node[below]{\S \ref{sec:blob-definition}} (BC);

\draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs);

\draw[->] (FU) -- node[right=10pt] {$\cF(M)/U$ \\ Defn \ref{defn:TQFT-invariant}} (A);

\draw[->] (tC) -- node[below] {Example \ref{ex:traditional-n-categories(fields)}\\ and \S \ref{sec:example:traditional-n-categories(fields)}} (FU);

\draw[->] (C.-100) -- node[left] {

	\S \ref{ss:ncat_fields}

	%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$

   } (FU.100);

\draw[->] (C.210) -- node[above left=3pt] {restrict to \\ standard balls} (tC.42);

\draw[->] (tC) -- node[below=4.5pt] {c.f. \S \ref{sec:comparing-defs}} (C.220);

\draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80);

\draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Proposition \ref{thm:skein-modules}} (A);

\draw[->] (FU) -- node[left] {blob complex \\ for balls \\ Example \ref{ex:blob-complexes-of-balls}} (Cs);

\draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs);

\end{tikzpicture}

\endpgfgraphicnamed%

\mbox{} % <-- weird, doesn't compile unless I put something here after the \endpgfgraphicnamed...? -S

}

\caption{The main gadgets and constructions of the paper.}

\label{fig:outline}

\end{figure}

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 cochains) in terms of the blob complex.

The appendices prove technical results about $\CH{M}$ 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. 

%Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, 

%thought of as a disk-like $n$-category, in terms of the topology of $M$.

\subsection{Motivation}

\label{sec:motivations}

We will briefly sketch our original motivation for defining the blob complex.

As a starting point, consider TQFTs constructed via fields and local relations.

(See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.)

This gives a satisfactory treatment for semisimple TQFTs

(i.e.\ TQFTs for which the cylinder 1-category associated to an

$n{-}1$-manifold $Y$ is semisimple for all $Y$).

For non-semi-simple TQFTs, this approach is less satisfactory.

Our main motivating example (though we will not develop it in this paper)

is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology.

It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together

with a link $L \subset \bd W$.

The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.

%\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}

How would we go about computing $A_{Kh}(W^4, L)$?

For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence)

relating resolutions of a crossing.

Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt

to compute $A_{Kh}(S^1\times B^3, L)$.

According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$

corresponds to taking a coend (self tensor product) over the cylinder category

associated to $B^3$ (with appropriate boundary conditions).

The coend is not an exact functor, so the exactness of the triangle breaks.

The obvious solution to this problem is to replace the coend with its derived counterpart, 

Hochschild homology.

This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology

of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.

If we build our manifold up via a handle decomposition, the computation

would be a sequence of derived coends.

A different handle decomposition of the same manifold would yield a different

sequence of derived coends.

To show that our definition in terms of derived coends is well-defined, we

would need to show that the above two sequences of derived coends yield 

isomorphic answers, and that the isomorphism does not depend on any

choices we made along the way.

This is probably not easy to do.

Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$

which is manifestly invariant.

(That is, a definition that does not

involve choosing a decomposition of $W$.

After all, one of the virtues of our starting point --- TQFTs via field and local relations ---

is that it has just this sort of manifest invariance.)

The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient

\[

 \text{linear combinations of fields} \;\big/\; \text{local relations} ,

\]

with an appropriately free resolution (the blob complex)

\[

	\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .

\]

Here $\bc_0$ is linear combinations of fields on $W$,

$\bc_1$ is linear combinations of local relations on $W$,

$\bc_2$ is linear combinations of relations amongst relations on $W$,

\subsection{Formal properties}

\label{sec:properties}

The blob complex enjoys the following list of formal properties.

\begin{property}[Functoriality]

\label{property:functoriality}%

The blob complex is functorial with respect to homeomorphisms.

That is, 

for a fixed $n$-dimensional system of fields $\cF$, the association

\begin{equation*}

X \mapsto \bc_*(X; \cF)

\end{equation*}

is a functor from $n$-manifolds and homeomorphisms between them to chain 

complexes and isomorphisms between them.

\end{property}

As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; 

this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below.

The blob complex is also functorial with respect to $\cF$, 

although we will not address this in detail here.

\begin{property}[Disjoint union]

\label{property:disjoint-union}

The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes.

\begin{equation*}

\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)

\end{equation*}

\end{property}

If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, 

write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.

Note that this includes the case of gluing two disjoint manifolds together.

\begin{property}[Gluing map]

\label{property:gluing-map}%

Given a gluing $X \to X_\mathrm{gl}$, there is an injective natural map

\[

	\bc_*(X) \to \bc_*(X_\mathrm{gl}) 

\]

(natural with respect to homeomorphisms, and also associative with respect to iterated gluings).

\end{property}

\begin{property}[Contractibility]

\label{property:contractibility}%

With field coefficients, the blob complex on an $n$-ball is contractible in the sense 

that it is homotopic to its $0$-th homology.

Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces 

associated by the system of fields $\cF$ to balls.

\begin{equation*}

\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)}

\end{equation*}

\end{property}

Property \ref{property:functoriality} will be immediate from the definition given in

\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.

Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and 

\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.

\subsection{Specializations}

\label{sec:specializations}

The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology.

\newtheorem*{thm:skein-modules}{Proposition \ref{thm:skein-modules}}

\begin{thm:skein-modules}[Skein modules]

The $0$-th blob homology of $X$ is the usual 

(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$

by $\cF$.

(See \S \ref{sec:local-relations}.)

\begin{equation*}

H_0(\bc_*(X;\cF)) \iso A_{\cF}(X)

\end{equation*}

\end{thm:skein-modules}

\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}}

\begin{thm:hochschild}[Hochschild homology when $X=S^1$]

The blob complex for a $1$-category $\cC$ on the circle is

quasi-isomorphic to the Hochschild complex.

\begin{equation*}

\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).}

\end{equation*}

\end{thm:hochschild}

Proposition \ref{thm:skein-modules} is immediate from the definition, and

Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}.

%We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of 

%certain commutative algebras thought of as $n$-categories.

\subsection{Structure of the blob complex}

\label{sec:structure}

In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.

\newtheorem*{thm:CH}{Theorem \ref{thm:CH}}

\begin{thm:CH}[$C_*(\Homeo(-))$ action]

There is a chain map

\begin{equation*}

e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).

\end{equation*}

such that

\begin{enumerate}

\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. 

\item For

any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram

(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy).

\begin{equation*}

\xymatrix@C+2cm{

     \CH{X} \otimes \bc_*(X)

        \ar[r]_{e_{X}}  \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &

            \bc_*(X) \ar[d]_{\gl_Y} \\

     \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow)

}

\end{equation*}

\end{enumerate}

%Moreover any such chain map is unique, up to an iterated homotopy.

%(That is, any pair of homotopies have a homotopy between them, and so on.)

%\nn{revisit this after proof below has stabilized}

\end{thm:CH}

\newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}}

Further,

\begin{thm:CH-associativity}

The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy).

\begin{equation*}

\xymatrix{

\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\

\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X)

}

\end{equation*}

\end{thm:CH-associativity}

Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps

$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$

for any homeomorphic pair $X$ and $Y$, 

satisfying corresponding conditions.

In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, 

from which we can construct systems of fields.

Below, when we talk about the blob complex for a disk-like $n$-category, 

we are implicitly passing first to this associated system of fields.

In that section we describe how to use the blob complex to 

\newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}}

%\label{thm:blobs-ainfty}

Let $Y$ be an $n{-}k$-manifold. 

to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set 

$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ 

(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) 

Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.

\end{ex:blob-complexes-of-balls}

\begin{rem}

Perhaps the most interesting case is when $Y$ is just a point; 

\end{rem}

instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}.

The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. 

%The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.

\newtheorem*{thm:product}{Theorem \ref{thm:product}}

\begin{thm:product}[Product formula]

Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.

Let $\cC$ be an $n$-category.

(see Example \ref{ex:blob-complexes-of-balls}).

Then

\[

	\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).

\]

\end{thm:product}

The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps

(see \S \ref{ss:product-formula}).

Fix a disk-like $n$-category $\cC$, which we'll omit from the notation.

\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}}

\begin{thm:gluing}[Gluing formula]

\mbox{}% <-- gets the indenting right

\begin{itemize}

\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an

$A_\infty$ module for $\bc_*(Y)$.

$\bc_*(X)$ as an $\bc_*(Y)$-bimodule:

\begin{equation*}

\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow

\end{equation*}

\end{itemize}

\end{thm:gluing}

Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.

\subsection{Applications}

\label{sec:applications}

Finally, we give two applications of the above machinery.

\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}}

\begin{thm:map-recon}[Mapping spaces]

$B^n \to T$.

(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)

Then 

\end{thm:map-recon}

This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. 

Note that there is no restriction on the connectivity of $T$.

The proof appears in \S \ref{sec:map-recon}.

\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}}

\begin{thm:deligne}[Higher dimensional Deligne conjecture]

The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains

(up to coherent homotopy).

Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad,

this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.

\end{thm:deligne}

See \S \ref{sec:deligne} for a full explanation of the statement, and the proof.