%!TEX root = ../blob1.tex

\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories}

\label{sec:ainfblob}

Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the 

anticlimactically tautological definition of the blob

complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.

We will show below 

in Corollary \ref{cor:new-old}

that when $\cC$ is obtained from a system of fields $\cD$ 

as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), 

$\cl{\cC}(M)$ is homotopy equivalent to

our original definition of the blob complex $\bc_*(M;\cD)$.

%\medskip

%An important technical tool in the proofs of this section is provided by the idea of ``small blobs".

%Fix $\cU$, an open cover of $M$.

%Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ 

%of all blob diagrams in which every blob is contained in some open set of $\cU$, 

%and moreover each field labeling a region cut out by the blobs is splittable 

%into fields on smaller regions, each of which is contained in some open set of $\cU$.

%

%\begin{thm}[Small blobs] \label{thm:small-blobs}

%The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence.

%\end{thm}

%The proof appears in \S \ref{appendix:small-blobs}.

\subsection{A product formula}

\label{ss:product-formula}

Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from 

Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $\cC_F$ 

defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and

$\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$.

\begin{thm} \label{thm:product}

Let $Y$ be a $k$-manifold.

Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) 

and ``new-fangled" (hocolimit) blob complexes

\[

	\cB_*(Y \times F) \htpy \cl{\cC_F}(Y) .

\]\end{thm}

\begin{proof}

We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}.

First we define a map 

\[

	\psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) .

\]

On 0-simplices of the hocolimit 

we just glue together the various blob diagrams on $X_i\times F$

(where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on

$Y\times F$.

For simplices of dimension 1 and higher we define the map to be zero.

It is easy to check that this is a chain map.

In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$

and a map

\[

	\phi: G_* \to \cl{\cC_F}(Y) .

\]

Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding

decomposition of $Y\times F$ into the pieces $X_i\times F$.

Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there

exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$.

It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ 

is homotopic to a subcomplex of $G_*$.

(If the blobs of $a$ are small with respect to a sufficiently fine cover then their

projections to $Y$ are contained in some disjoint union of balls.)

Note that the image of $\psi$ is equal to $G_*$.

We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models.

Let $a$ be a generator of $G_*$.

Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$

such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing

in an iterated boundary of $a$ (this includes $a$ itself).

(Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;

see \S\ref{ss:ncat_fields}.)

By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is 

$b$ split according to $K_0\times F$.

To simplify notation we will just write plain $b$ instead of $b^\sharp$.

Roughly speaking, $D(a)$ consists of 0-simplices which glue up to give

$a$ (or one of its iterated boundaries), 1-simplices which connect all the 0-simplices, 

2-simplices which kill the homology created by the 

1-simplices, and so on.

More formally,

\begin{lemma} \label{lem:d-a-acyclic}

$D(a)$ is acyclic.

\end{lemma}

\begin{proof}

We will prove acyclicity in the first couple of degrees, and 

%\nn{in this draft, at least}

leave the general case to the reader.

Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$.

We want to find 1-simplices which connect $K$ and $K'$.

We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily

the case.

(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.)

However, we {\it can} find another decomposition $L$ such that $L$ shares common

refinements with both $K$ and $K'$.

Let $KL$ and $K'L$ denote these two refinements.

Then 1-simplices associated to the four anti-refinements

$KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$

give the desired chain connecting $(a, K)$ and $(a, K')$

(see Figure \ref{zzz4}).

\begin{figure}[t] \centering

\begin{tikzpicture}

\foreach \x/\label in {-3/K, 0/L, 3/K'} {

	\node(\label) at (\x,0) {$\label$};

}

\foreach \x/\la/\lb in {-1.5/K/L, 1.5/K'/L} {

	\node(\la \lb) at (\x,-1.5) {$\la \lb$};

	\draw[->] (\la \lb) -- (\la);

	\draw[->] (\la \lb) -- (\lb); 

}

\end{tikzpicture}

\caption{Connecting $K$ and $K'$ via $L$}

\label{zzz4}

\end{figure}

Consider a different choice of decomposition $L'$ in place of $L$ above.

This leads to a cycle of 1-simplices.

We want to find 2-simplices which fill in this cycle.

Choose a decomposition $M$ which has common refinements with each of 

$K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.

(We also also require that $KLM$ antirefines to $KM$, etc.)

Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick.

(Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.)

\begin{figure}[t] \centering

\begin{tikzpicture}

\node(M) at (0,0) {$M$};

\foreach \angle/\label in {0/K', 45/K'L, 90/L, 135/KL, 180/K, 225/KL', 270/L', 315/K'L'} {

	\node(\label) at (\angle:4) {$\label$};

}

\foreach \label in {K', L, K, L'} {

	\node(\label M) at ($(M)!0.6!(\label)$) {$\label M$};

	\draw[->] (\label M)--(M);

	\draw[->] (\label M)--(\label);

}

\foreach \k in {K, K'} {

	\foreach \l in {L, L'} {

		\node(\k \l M) at (intersection cs: first line={(\k M)--(\l)}, second line={(\l M)--(\k)}) {$\k \l M$};

		\draw[->] (\k \l M)--(M);

		\draw[->] (\k \l M)--(\k \l );

		\draw[->] (\k \l M)--(\k M);

		\draw[->] (\k \l M)--(\l);

		\draw[->] (\k \l M)--(\l M);

		\draw[->] (\k \l M)--(\k);

	}

}

\draw[->] (K'L') to[bend right=10] (K');

\draw[->] (K'L') to[bend left=10] (L');

\draw[->] (KL') to[bend left=10] (K);

\draw[->] (KL') to[bend right=10] (L');

\draw[->] (K'L) to[bend left=10] (K');

\draw[->] (K'L) to[bend right=10] (L);

\draw[->] (KL) to[bend right=10] (K);

\draw[->] (KL) to[bend left=10] (L);

\end{tikzpicture}

\caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$}

\label{zzz5}

\end{figure}

Continuing in this way we see that $D(a)$ is acyclic.

\end{proof}

We are now in a position to apply the method of acyclic models to get a map

$\phi:G_* \to \cl{\cC_F}(Y)$.

We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex

and $r$ is a sum of simplices of dimension 1 or higher.

We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.

First, $\psi\circ\phi$ is the identity on the nose:

\[

	\psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0.

\]

Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and 

$\psi$ glues those pieces back together, yielding $a$.

We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices.

Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.

To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.

Both the identity map and $\phi\circ\psi$ are compatible with this

collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps

are homotopic.

This concludes the proof of Theorem \ref{thm:product}.

\end{proof}

\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}

\medskip

Taking $F$ above to be a point, we obtain the following corollary.

\begin{cor}

\label{cor:new-old}

Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$

$n$-category obtained from $\cE$ by taking the blob complex of balls.

Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are

homotopy equivalent:

\[

	\bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) .

\]

\end{cor}

\medskip

Theorem \ref{thm:product} extends to the case of general fiber bundles

\[

	F \to E \to Y ,

\]

an indeed even to the case of general maps

\[

	M\to Y .

\]

We outline two approaches to these generalizations.

The first is somewhat tautological, while the second is more amenable to

calculation.

We can generalize the definition of a $k$-category by replacing the categories

of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$

(c.f. \cite{MR2079378}).

Call this a $k$-category over $Y$.

A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:

assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$,

or the fields $\cE(p^*(E))$, if $\dim(D) < k$.

($p^*(E)$ denotes the pull-back bundle over $D$.)

Let $\cF_E$ denote this $k$-category over $Y$.

We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to

get a chain complex $\cl{\cF_E}(Y)$.

The proof of Theorem \ref{thm:product} goes through essentially unchanged 

to show that

\begin{thm}

Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.

Then

\[

	\bc_*(E) \simeq \cl{\cF_E}(Y) .

\]

\qed

\end{thm}

We can generalize this result still further by noting that it is not really necessary

for the definition of $\cF_E$ that $E\to Y$ be a fiber bundle.

Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$.

Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product

$D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$.

(If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$

lying above $D$.)

We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which a re good with respect to $M$.

We can again adapt the homotopy colimit construction to

get a chain complex $\cl{\cF_M}(Y)$.

The proof of Theorem \ref{thm:product} again goes through essentially unchanged 

to show that

\begin{thm}

Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above.

Then

\[

	\bc_*(M) \simeq \cl{\cF_M}(Y) .

\]

\qed

\end{thm}

\medskip

In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) 

and various sphere modules based on $F \to E \to Y$

or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$.

Information about the specific map to $Y$ has been taken out of the categories

and put into sphere modules and decorations.

%Let $F \to E \to Y$ be a fiber bundle as above.

%Choose a decomposition $Y = \cup X_i$

%such that the restriction of $E$ to $X_i$ is a product $F\times X_i$,

%and choose trivializations of these products as well.

%

%\nn{edit marker}

%To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module).

%And more generally to each codim-$j$ face we have an $S^{j-1}$-module.

%Decorate the decomposition with these modules and do the colimit.

%

%

%\nn{There is a version of this last construction for arbitrary maps $E \to Y$

%(not necessarily a fibration).}

%

%

%

%Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one.

%Let $X_1$ and $X_2$ be $n$-manifolds

%

\subsection{A gluing theorem}

\label{sec:gluing}

Next we prove a gluing theorem.

Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.

We will need an explicit collar on $Y$, so rewrite this as

$X = X_1\cup (Y\times J) \cup X_2$.

Given this data we have:

\begin{itemize}

\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball

$D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$

(for $m+k = n$).

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

%\nn{need to explain $c$}.

\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.

\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked

$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)

or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).

(See Example \ref{bc-module-example}.)

\end{itemize}

\nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe

just say that until we define functors we can't do more}

\begin{thm}

\label{thm:gluing}

$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.

\end{thm}

\begin{proof}

We will assume $k=n$; the other cases are similar.

The proof is similar to that of Theorem \ref{thm:product}.

We give a short sketch with emphasis on the differences from 

the proof of Theorem \ref{thm:product}.

Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.

Recall that this is a homotopy colimit based on decompositions of the interval $J$.

We define a map $\psi:\cT\to \bc_*(X)$.

On 0-simplices it is given

by gluing the pieces together to get a blob diagram on $X$.

On simplices of dimension 1 and greater $\psi$ is zero.

The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split

over some decomposition of $J$.

It follows from Lemma \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to 

a subcomplex of $G_*$. 

Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.

As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$

an acyclic subcomplex which is (roughly) $\psi\inv(a)$.

The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have

a common refinement.

The proof that these two maps are inverse to each other is the same as in

Theorem \ref{thm:product}.

\end{proof}

\medskip

\subsection{Reconstructing mapping spaces}

\label{sec:map-recon}

The next theorem shows how to reconstruct a mapping space from local data.

Let $T$ be a topological space, let $M$ be an $n$-manifold, 

and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ 

of Example \ref{ex:chains-of-maps-to-a-space}.

Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever

want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).

To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.

\begin{thm}

\label{thm:map-recon}

The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ 

is quasi-isomorphic to singular chains on maps from $M$ to $T$.

$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$

\end{thm}

\begin{rem}

Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology 

of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers 

the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected.

This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} 

that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which 

is trivial at levels 0 through $n-1$.

Ricardo Andrade also told us about a similar result.

\end{rem}

\begin{proof}

The proof is again similar to that of Theorem \ref{thm:product}.

We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.

Recall that 

the 0-simplices of the homotopy colimit $\cB^\cT(M)$ 

are a direct sum of chain complexes with the summands indexed by

decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms

of $\cT$.

Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs

$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous

map from the $n{-}1$-skeleton of $K$ to $T$.

The summand indexed by $(K, \vphi)$ is

\[

	\bigotimes_b D_*(b, \vphi),

\]

where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes

chains of maps from $b$ to $T$ compatible with $\vphi$.

We can take the product of these chains of maps to get chains of maps from

all of $M$ to $K$.

This defines $\psi$ on 0-simplices.

We define $\psi$ to be zero on $(\ge1)$-simplices.

It is not hard to see that this defines a chain map from 

$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.

The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by 

families of maps whose support is contained in a disjoint union of balls.

It follows from Lemma \ref{extension_lemma_c} 

that $C_*(\Maps(M\to T))$ is homotopic to a subcomplex of $G_*$.

We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models.

Let $a$ be a generator of $G_*$.

Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all 

pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$

and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$.

(See the proof of Theorem \ref{thm:product} for more details.)

The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic.

By the usual acyclic models nonsense, there is a (unique up to homotopy)

map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.

Furthermore, we may choose $\phi$ such that for all $a$ 

\[

	\phi(a) = (a, K) + r

\]

where $(a, K)$ is a 0-simplex and $r$ is a sum of simplices of dimension 1 and greater.

It is now easy to see that $\psi\circ\phi$ is the identity on the nose.

Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.

(See the proof of Theorem \ref{thm:product} for more details.)