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\title{Blob Homology}

\begin{document}



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\textbf{Draft version, do not distribute.}

%\versioninfo
[later than 11 June 2009]

\noop{

\section*{Todo}

\subsection*{What else?...}

\begin{itemize}
\item higher priority
\begin{itemize}
\item K\&S: learn the state of the art in A-inf categories
(tensor products, Kadeishvili result, ...)
\item K: so-called evaluation map stuff
\item K: topological fields
\item section describing intended applications
\item say something about starting with semisimple n-cat (trivial?? not trivial?)
\item T.O.C.
\end{itemize}
\item medium priority
\begin{itemize}
\item $n=2$ examples
\item dimension $n+1$ (generalized Deligne conjecture?)
\item should be clear about PL vs Diff; probably PL is better
(or maybe not)
\item something about higher derived coend things (derived 2-coend, e.g.)
\item shuffle product vs gluing product (?)
\item connection between $A_\infty$ operad and topological $A_\infty$ cat defs
\end{itemize}
\item lower priority
\begin{itemize}
\item Derive Hochschild standard results from blob point of view?
\item Kh
\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations.
\end{itemize}
\end{itemize}

} %end \noop



\input{text/intro.tex}


\section{Definitions}
\label{sec:definitions}

\subsection{Systems of fields}
\label{sec:fields}

Let $\cM_k$ denote the category (groupoid, in fact) with objects 
oriented PL manifolds of dimension
$k$ and morphisms homeomorphisms.
(We could equally well work with a different category of manifolds ---
unoriented, topological, smooth, spin, etc. --- but for definiteness we
will stick with oriented PL.)

Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.

A $n$-dimensional {\it system of fields} in $\cS$
is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
together with some additional data and satisfying some additional conditions, all specified below.

\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris}

Before finishing the definition of fields, we give two motivating examples
(actually, families of examples) of systems of fields.

The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
from X to $B$.

The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be 
the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
$j$-morphisms of $C$.
One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
This is described in more detail below.

Now for the rest of the definition of system of fields.
\begin{enumerate}
\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, 
and these maps are a natural
transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.
For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of 
$\cC(X)$ which restricts to $c$.
In this context, we will call $c$ a boundary condition.
\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$.
\item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps
again comprise a natural transformation of functors.
In addition, the orientation reversal maps are compatible with the boundary restriction maps.
\item $\cC_k$ is compatible with the symmetric monoidal
structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
compatibly with homeomorphisms, restriction to boundary, and orientation reversal.
We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
restriction maps.
\item Gluing without corners.
Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
Using the boundary restriction, disjoint union, and (in one case) orientation reversal
maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two
copies of $Y$ in $\bd X$.
Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps.
Then (here's the axiom/definition part) there is an injective ``gluing" map
\[
	\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,
\]
and this gluing map is compatible with all of the above structure (actions
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
the gluing map is surjective.
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the 
gluing surface, we say that fields in the image of the gluing map
are transverse to $Y$ or cuttable along $Y$.
\item Gluing with corners.
Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries.
Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
(without corners) along two copies of $\bd Y$.
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let
$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.
Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.
(This restriction map uses the gluing without corners map above.)
Using the boundary restriction, gluing without corners, and (in one case) orientation reversal
maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
copies of $Y$ in $\bd X$.
Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps.
Then (here's the axiom/definition part) there is an injective ``gluing" map
\[
	\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) ,
\]
and this gluing map is compatible with all of the above structure (actions
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
the gluing map is surjective.
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the 
gluing surface, we say that fields in the image of the gluing map
are transverse to $Y$ or cuttable along $Y$.
\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
$c \mapsto c\times I$.
These maps comprise a natural transformation of functors, and commute appropriately
with all the structure maps above (disjoint union, boundary restriction, etc.).
Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.
\end{enumerate}

\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$}

\bigskip
Using the functoriality and $\bullet\times I$ properties above, together
with boundary collar homeomorphisms of manifolds, we can define the notion of 
{\it extended isotopy}.
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
of $\bd M$.
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$.
Let $c$ be $x$ restricted to $Y$.
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$.
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$.
More generally, we define extended isotopy to be the equivalence relation on fields
on $M$ generated by isotopy plus all instance of the above construction
(for all appropriate $Y$ and $x$).

\nn{should also say something about pseudo-isotopy}

%\bigskip
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\nn{note: probably will suppress from notation the distinction
between fields and their (orientation-reversal) duals}

\nn{remark that if top dimensional fields are not already linear
then we will soon linearize them(?)}

We now describe in more detail systems of fields coming from sub-cell-complexes labeled
by $n$-category morphisms.

Given an $n$-category $C$ with the right sort of duality
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
we can construct a system of fields as follows.
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
with codimension $i$ cells labeled by $i$-morphisms of $C$.
We'll spell this out for $n=1,2$ and then describe the general case.

If $X$ has boundary, we require that the cell decompositions are in general
position with respect to the boundary --- the boundary intersects each cell
transversely, so cells meeting the boundary are mere half-cells.

Put another way, the cell decompositions we consider are dual to standard cell
decompositions of $X$.

We will always assume that our $n$-categories have linear $n$-morphisms.

For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
an object (0-morphism) of the 1-category $C$.
A field on a 1-manifold $S$ consists of
\begin{itemize}
    \item A cell decomposition of $S$ (equivalently, a finite collection
of points in the interior of $S$);
    \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
by an object (0-morphism) of $C$;
    \item a transverse orientation of each 0-cell, thought of as a choice of
``domain" and ``range" for the two adjacent 1-cells; and
    \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
domain and range determined by the transverse orientation and the labelings of the 1-cells.
\end{itemize}

If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
of the algebra.

\medskip

For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories)
that are common in the literature.
We describe these carefully here.

A field on a 0-manifold $P$ is a labeling of each point of $P$ with
an object of the 2-category $C$.
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
A field on a 2-manifold $Y$ consists of
\begin{itemize}
    \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
that each component of the complement is homeomorphic to a disk);
    \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
by a 0-morphism of $C$;
    \item a transverse orientation of each 1-cell, thought of as a choice of
``domain" and ``range" for the two adjacent 2-cells;
    \item a labeling of each 1-cell by a 1-morphism of $C$, with
domain and range determined by the transverse orientation of the 1-cell
and the labelings of the 2-cells;
    \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
to $\pm 1 \in S^1$; and
    \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
determined by the labelings of the 1-cells and the parameterizations of the previous
bullet.
\end{itemize}
\nn{need to say this better; don't try to fit everything into the bulleted list}

For general $n$, a field on a $k$-manifold $X^k$ consists of
\begin{itemize}
    \item A cell decomposition of $X$;
    \item an explicit general position homeomorphism from the link of each $j$-cell
to the boundary of the standard $(k-j)$-dimensional bihedron; and
    \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
domain and range determined by the labelings of the link of $j$-cell.
\end{itemize}

%\nn{next definition might need some work; I think linearity relations should
%be treated differently (segregated) from other local relations, but I'm not sure
%the next definition is the best way to do it}

\medskip

For top dimensional ($n$-dimensional) manifolds, we're actually interested
in the linearized space of fields.
By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
the vector space of finite
linear combinations of fields on $X$.
If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
Thus the restriction (to boundary) maps are well defined because we never
take linear combinations of fields with differing boundary conditions.

In some cases we don't linearize the default way; instead we take the
spaces $\lf(X; a)$ to be part of the data for the system of fields.
In particular, for fields based on linear $n$-category pictures we linearize as follows.
Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
obvious relations on 0-cell labels.
More specifically, let $L$ be a cell decomposition of $X$
and let $p$ be a 0-cell of $L$.
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
Then the subspace $K$ is generated by things of the form
$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
to infer the meaning of $\alpha_{\lambda c + d}$.
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.

\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
will do something similar below; in general, whenever a label lives in a linear
space we do something like this; ? say something about tensor
product of all the linear label spaces?  Yes:}

For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
Define an ``almost-field" to be a field without labels on the 0-cells.
(Recall that 0-cells are labeled by $n$-morphisms.)
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
space determined by the labeling of the link of the 0-cell.
(If the 0-cell were labeled, the label would live in this space.)
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
above tensor products.



\subsection{Local relations}
\label{sec:local-relations}


A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
for all $n$-manifolds $B$ which are
homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, 
satisfying the following properties.
\begin{enumerate}
\item functoriality: 
$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
\item local relations imply extended isotopy: 
if $x, y \in \cC(B; c)$ and $x$ is extended isotopic 
to $y$, then $x-y \in U(B; c)$.
\item ideal with respect to gluing:
if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
\end{enumerate}
See \cite{kw:tqft} for details.


For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$,
where $a$ and $b$ are maps (fields) which are homotopic rel boundary.

For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
domain and range.

\nn{maybe examples of local relations before general def?}

Given a system of fields and local relations, we define the skein space
$A(Y^n; c)$ to be the space of all finite linear combinations of fields on
the $n$-manifold $Y$ modulo local relations.
The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
is defined to be the dual of $A(Y; c)$.
(See \cite{kw:tqft} or xxxx for details.)

\nn{should expand above paragraph}

The blob complex is in some sense the derived version of $A(Y; c)$.



\subsection{The blob complex}
\label{sec:blob-definition}

Let $X$ be an $n$-manifold.
Assume a fixed system of fields and local relations.
In this section we will usually suppress boundary conditions on $X$ from the notation
(e.g. write $\lf(X)$ instead of $\lf(X; c)$).

We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
submanifold of $X$, then $X \setmin Y$ implicitly means the closure
$\overline{X \setmin Y}$.

We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.

Define $\bc_0(X) = \lf(X)$.
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
We'll omit this sort of detail in the rest of this section.)
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.

$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.
Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear
combinations of 1-blob diagrams, where a 1-blob diagram to consists of
\begin{itemize}
\item An embedded closed ball (``blob") $B \sub X$.
\item A field $r \in \cC(X \setmin B; c)$
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
\item A local relation field $u \in U(B; c)$
(same $c$ as previous bullet).
\end{itemize}
In order to get the linear structure correct, we (officially) define
\[
	\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
\]
The first direct sum is indexed by all blobs $B\subset X$, and the second
by all boundary conditions $c \in \cC(\bd B)$.
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$.

Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by 
\[ 
	(B, u, r) \mapsto u\bullet r, 
\]
where $u\bullet r$ denotes the linear
combination of fields on $X$ obtained by gluing $u$ to $r$.
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
just erasing the blob from the picture
(but keeping the blob label $u$).

Note that the skein space $A(X)$
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.

$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the 
local relations encoded in $\bc_1(X)$.
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
2-blob diagrams, of which there are two types, disjoint and nested.

A disjoint 2-blob diagram consists of
\begin{itemize}
\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
(where $c_i \in \cC(\bd B_i)$).
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$.
\end{itemize}
We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
reversing the order of the blobs changes the sign.
Define $\bd(B_0, B_1, u_0, u_1, r) = 
(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
In other words, the boundary of a disjoint 2-blob diagram
is the sum (with alternating signs)
of the two ways of erasing one of the blobs.
It's easy to check that $\bd^2 = 0$.

A nested 2-blob diagram consists of
\begin{itemize}
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
\item A field $r \in \cC(X \setmin B_0; c_0)$
(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$.
\item A local relation field $u_0 \in U(B_0; c_0)$.
\end{itemize}
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
(for some $c_1 \in \cC(B_1)$) and
$r' \in \cC(X \setmin B_1; c_1)$.
Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
Note that the requirement that
local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
sum of the two ways of erasing one of the blobs.
If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
It is again easy to check that $\bd^2 = 0$.

\nn{should draw figures for 1, 2 and $k$-blob diagrams}

As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
(officially)
\begin{eqnarray*}
	\bc_2(X) & \deq &
	\left( 
		\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}
			U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)
	\right) \\
	&& \bigoplus \left( 
		\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0}
			U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)
	\right) .
\end{eqnarray*}
The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$,
but we didn't feel like introducing a notation for that.
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
(rather than a new, linearly independent 2-blob diagram).

Now for the general case.
A $k$-blob diagram consists of
\begin{itemize}
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or
$B_i \sub B_j$ or $B_j \sub B_i$.
(The case $B_i = B_j$ is allowed.
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
If a blob has no other blobs strictly contained in it, we call it a twig blob.
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
(These are implied by the data in the next bullets, so we usually
suppress them from the notation.)
$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
if the latter space is not empty.
\item A field $r \in \cC(X \setmin B^t; c^t)$,
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
is determined by the $c_i$'s.
$r$ is required to be cuttable along the boundaries of all blobs, twigs or not.
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
If $B_i = B_j$ then $u_i = u_j$.
\end{itemize}

If two blob diagrams $D_1$ and $D_2$ 
differ only by a reordering of the blobs, then we identify
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.

$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
As before, the official definition is in terms of direct sums
of tensor products:
\[
	\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
		\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
\]
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
$\overline{c}$ runs over all boundary conditions, again as described above.
$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$.

The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
Let $E_j(b)$ denote the result of erasing the $j$-th blob.
If $B_j$ is not a twig blob, this involves only decrementing
the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
If $B_j$ is a twig blob, we have to assign new local relation labels
if removing $B_j$ creates new twig blobs.
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
Finally, define
\eq{
    \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
}
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
Thus we have a chain complex.

\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}

\nn{?? remark about dendroidal sets}



\section{Basic properties of the blob complex}
\label{sec:basic-properties}

\begin{prop} \label{disjunion}
There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
\end{prop}
\begin{proof}
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
blob diagram $(b_1, b_2)$ on $X \du Y$.
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
a pair of blob diagrams on $X$ and $Y$.
These two maps are compatible with our sign conventions.
The two maps are inverses of each other.
\nn{should probably say something about sign conventions for the differential
in a tensor product of chain complexes; ask Scott}
\end{proof}

For the next proposition we will temporarily restore $n$-manifold boundary
conditions to the notation.

Suppose that for all $c \in \cC(\bd B^n)$
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
of the quotient map
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
For example, this is always the case if you coefficient ring is a field.
Then
\begin{prop} \label{bcontract}
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$
is a chain homotopy equivalence
with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.
Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0.
\end{prop}
\begin{proof}
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map
$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$.
For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding
an $(i{+}1)$-st blob equal to all of $B^n$.
In other words, add a new outermost blob which encloses all of the others.
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
\end{proof}

Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.

For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$,
where $(c', c'')$ is some (any) splitting of $c$ into domain and range.

\medskip

\nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction.
But I think it's worth saying that the Diff actions will be enhanced later.
Maybe put that in the intro too.}

As we noted above,
\begin{prop}
There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.
\qed
\end{prop}


\begin{prop}
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
of $n$-manifolds and diffeomorphisms to the category of chain complexes and
(chain map) isomorphisms.
\qed
\end{prop}

In particular,
\begin{prop}  \label{diff0prop}
There is an action of $\Diff(X)$ on $\bc_*(X)$.
\qed
\end{prop}

The above will be greatly strengthened in Section \ref{sec:evaluation}.

\medskip

For the next proposition we will temporarily restore $n$-manifold boundary
conditions to the notation.

Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$.
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$
with boundary $Z\sgl$.
Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
we have the blob complex $\bc_*(X; a, b, c)$.
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
$X$ to get blob diagrams on $X\sgl$:

\begin{prop}
There is a natural chain map
\eq{
    \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
}
The sum is over all fields $a$ on $Y$ compatible at their
($n{-}2$-dimensional) boundaries with $c$.
`Natural' means natural with respect to the actions of diffeomorphisms.
\qed
\end{prop}

The above map is very far from being an isomorphism, even on homology.
This will be fixed in Section \ref{sec:gluing} below.

\nn{Next para not need, since we already use bullet = gluing notation above(?)}

An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
and $X\sgl = X_1 \cup_Y X_2$.
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
For $x_i \in \bc_*(X_i)$, we introduce the notation
\eq{
    x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
}
Note that we have resumed our habit of omitting boundary labels from the notation.





\section{Hochschild homology when $n=1$}
\label{sec:hochschild}
\input{text/hochschild}




\section{Action of $\CD{X}$}
\label{sec:evaluation}
\input{text/evmap}



\input{text/ncat.tex}

\input{text/A-infty.tex}

\input{text/gluing.tex}



\section{Commutative algebras as $n$-categories}

\nn{this should probably not be a section by itself.  i'm just trying to write down the outline 
while it's still fresh in my mind.}

If $C$ is a commutative algebra it
can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for
$j<n$ and whose $n$-morphisms are $C$. 
The goal of this \nn{subsection?} is to compute
$\bc_*(M^n, C)$ for various commutative algebras $C$.

Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$.

Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$
unlabeled points in $M$.
Note that $\Sigma^0(M)$ is a point.
Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$.

Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$.

\begin{prop} \label{sympowerprop}
$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$.
\end{prop}

\begin{proof}
To define the chain maps between the two complexes we will use the following lemma:

\begin{lemma}
Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with
a basis (e.g.\ blob diagrams or singular simplices).
For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$
such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.
Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that
$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).
\end{lemma}

\begin{proof}
\nn{easy, but should probably write the details eventually}
\end{proof}

Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$
satisfying the conditions of the above lemma.
If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a 
finite unordered collection of points of $M$ with multiplicities, which is
a point in $\Sigma^\infty(M)$.
Define $R(b)_*$ to be the singular chain complex of this point.
If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs).
The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed 
by the numbers of points in each component of $D$.
We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so
$u$ picks out a component $X \sub \Sigma^\infty(D)$.
The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$,
and using this point we can embed $X$ in $\Sigma^\infty(M)$.
Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a 
subspace of $\Sigma^\infty(M)$.
It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma.
Thus we have defined (up to homotopy) a map from 
$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$.

Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace
$R(c)_* \sub \bc_*(M^n, k[t])$.
If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and 
$\Sigma^\infty(M)$ described above.
Now let $c$ be an $i$-simplex of $\Sigma^j(M)$.
Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$.
We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$
is homotopy equivalent to the subcomplex of small simplices.
How small?  $(2r)/3j$, where $r$ is the radius of injectivity of the metric.
Let $T\sub M$ be the ``track" of $c$ in $M$.
\nn{do we need to define this precisely?}
Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter.
\nn{need to say more precisely how small}
Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$.
This is contractible by \ref{bcontract}.
We can arrange that the boundary/inclusion condition is satisfied if we start with
low-dimensional simplices and work our way up.
\nn{need to be more precise}

\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity
(for this, might need a lemma that says we can assume that blob diameters are small)}
\end{proof}


\begin{prop} \label{ktcdprop}
The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$.
\end{prop}

\begin{proof}
The actions agree in degree 0, and both are compatible with gluing.
(cf. uniqueness statement in \ref{CDprop}.)
\end{proof}

\medskip

In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$,
and that the cyclic homology of $k[t]$ is related to the action of rotations
on $C_*(\Sigma^\infty(S^1), k)$.
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
Let us check this directly.

According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$.
\nn{say something about $t$-degree?  is this in [Loday]?}

We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other.
The fixed points of this flow are the equally spaced configurations.
This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation).
The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, 
and the holonomy of the $\Delta^{j-1}$ bundle
over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices.

In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and
of course $\Sigma^0(S^1)$ is a point.
Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$
and is zero for $i\ge 2$.
\nn{say something about $t$-degrees also matching up?}

By xxxx and \ref{ktcdprop}, 
the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$.
Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$.
If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree
0, $\z/j \z$ in odd degrees, and 0 in positive even degrees.
The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even 
degrees and 0 in odd degrees.
This agrees with the calculation in \nn{Loday, 3.1.7}.

\medskip

Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables.
Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations
of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable.
The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers
corresponding to the number of points of each color of a configuration.
A proof similar to that of \ref{sympowerprop} shows that

\begin{prop}
$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$.
\end{prop}

According to \nn{Loday, 3.2.2},
\[
	HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] .
\]
Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$.
We will content ourselves with the case $k = \z$.
One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact.
This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent
to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple
corresponding to $X$.
The homology calculation we desire follows easily from this.

\nn{say something about cyclic homology in this case?  probably not necessary.}

\medskip

Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$.
Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or
more points coinciding.

\begin{prop}
$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$
(relative singular chains with coefficients in $k$).
\end{prop}

\begin{proof}
\nn{...}
\end{proof}

\nn{...}




\appendix

\input{text/famodiff.tex}

\section{Comparing definitions of $A_\infty$ algebras}
\label{sec:comparing-A-infty}
In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.

We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as:
\begin{defn}
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
\begin{itemize}
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions
\item and a compatible action of $\CD{[0,1]}$.
\end{itemize}
\end{defn}
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions.
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).

Translating between these definitions is straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$
where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?}

%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$.  Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$.

%\begin{defn}
%A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'.

%A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity 
%\begin{equation*}
%f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}).
%\end{equation*}

%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that 
%\begin{enumerate}
%\item The diagram 
%\begin{equation*}
%\xymatrix{
%\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A
%}
%\end{equation*}
%commutes up to weakly unique homotopy.
%\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that
%\begin{equation*}
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)).
%\end{equation*}
%\end{enumerate}
%\end{defn}

From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
\begin{equation*}
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)).
\end{equation*}

It's then easy to calculate that
\begin{align*}
d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\
 & = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\
 & = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\
\intertext{and thus that}
m_1 \circ m_3 & =  m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1)
\end{align*}
as required (c.f. \cite[p. 6]{MR1854636}).
\todo{then the general case.}
We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.

\section{Morphisms and duals of topological $A_\infty$ modules}
\label{sec:A-infty-hom-and-duals}%

\begin{defn}
If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that  for each interval $J'$ the diagram
\begin{equation*}
\xymatrix{
\cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\
\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}}                                & \cN(J' cup J,a) 
}
\end{equation*}
commutes on the nose, and the diagram
\begin{equation*}
\xymatrix{
\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\
\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}}  & \cN(J',p';a) \\
}
\end{equation*}
commutes up to a weakly unique homotopy.
\end{defn}

The variations required for right modules and bimodules should be obvious.

\todo{duals}
\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic}


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This paper is available online at \arxiv{?????}, and at
\url{http://tqft.net/blobs},
and at \url{http://canyon23.net/math/}.

% A GTART necessity:
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%Recall that for $n$-category picture fields there is an evaluation map
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$.
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$.