%!TEX root = ../blob1.tex

\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 generalisation of several well-understood 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 $\cC$. (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}.)

\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}).

\subsubsection{Structure of the paper}

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 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.

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 $X$.

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.

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

view and using chain complexes in many places where we could be 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}

\medskip\hrule\medskip

[Old outline for intro]

\begin{itemize}

\item Starting point: TQFTs via fields and local relations.

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 $Y$).

\item For non-semiemple TQFTs, this approach is less satisfactory.

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

is the $4{+}1$-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 $A_{Kh}(B^4, L)$.

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

For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)

\nn{... $L_1, L_2, L_3$}.

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-via-fields, 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.

\item The obvious solution to this problem is to replace the coend with its derived counterpart.

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 the same answer.

This is probably not easy to do.

\item 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.)

\item 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 $W$,

and so on.

\item None of the above ideas depend on the details of the Khovanov homology example,

so we develop the general theory in the paper and postpone specific applications

to later papers.

\end{itemize}

\subsection{Formal properties}

\label{sec:properties}

\begin{property}[Functoriality]

\label{property:functoriality}%

The blob complex is functorial with respect to homeomorphisms. That is, 

for fixed $n$-category / fields $\cC$, the association

\begin{equation*}

X \mapsto \bc_*^{\cC}(X)

\end{equation*}

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

\end{property}

The blob complex is also functorial with respect to $\cC$, 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 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 $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$-submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ 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}%

%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map

%\begin{equation*}

%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).

%\end{equation*}

Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is

a natural map

\[

	\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) .

\]

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

\end{property}

\begin{property}[Contractibility]

\label{property:contractibility}%

\todo{Err, requires a splitting?}

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))}

\end{equation}

\end{property}

\begin{property}[Skein modules]

\label{property: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 $\cC$. (See \S \ref{sec:local-relations}.)

\begin{equation*}

H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)

\end{equation*}

\end{property}

\begin{property}[Hochschild homology when $X=S^1$]

\label{property:hochschild}%

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

quasi-isomorphic to the Hochschild complex.

\begin{equation*}

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

\end{equation*}

\end{property}

Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.

\begin{property}[$C_*(\Diff(-))$ action]

\label{property:evaluation}%

There is a chain map

\begin{equation*}

\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X).

\end{equation*}

Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for

any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram

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

\begin{equation*}

\xymatrix{

     \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X}    & \bc_*(X) \\

     \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2)

        \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}}  \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y}  &

            \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y}

}

\end{equation*}

\nn{should probably say something about associativity here (or not?)}

\nn{maybe do self-gluing instead of 2 pieces case:}

Further, 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.

\begin{equation*}

\xymatrix@C+2cm{

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

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

        \ar[r]_{\ev_{X}}  \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y}  &

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

}

\end{equation*}

\end{property}

There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category

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

\begin{property}[Product formula]

Then

\[

	\bc_*(Y^{n-k}\times W^k, \cC) \simeq \bc_*(W, A_*(Y)) .

\]

Note on the right here we have the version of the blob complex for $A_\infty$ $n$-categories.

\end{property}

\begin{property}[Gluing formula]

\label{property:gluing}%

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

\begin{itemize}

\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is

naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.

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

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

\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of

$\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule.

\begin{equation*}

\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow

\end{equation*}

\end{itemize}

\end{property}

\begin{property}[Mapping spaces]

Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps 

$B^n \to W$.

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

Then 

$$\bc_*(M, \pi^\infty_{\le n}(W) \simeq \CM{M}{W}.$$

\end{property}

\begin{property}[Higher dimensional Deligne conjecture]

The singular chains of the $n$-dimensional fat graph operad act on blob cochains.

\end{property}

See \S \ref{sec:deligne} for an explanation of the terms appearing here, and (in a later edition of this paper) the proof.

Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in

\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.}

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

Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation},

and Property \ref{property:gluing} in \S \ref{sec:gluing}.

\nn{need to say where the remaining properties are proved.}

\subsection{Future directions}

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.

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.

Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh}

\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.