\subsection{Topological $A_\infty$ categories}

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 $\cJ_1 \circ_m \cJ_2$.

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 `action of families of diffeomorphisms'.

A \emph{composition map} is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^k \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity 

\subsection{Homological systems of fields}

\subsection{Blob homology}

The definition of blob homology for $(\cF, \cU)$ a homological system of fields and local relations is essentially the same as that given before in \S \ref{???}.

The blob complex $\cB_*^{\cF,\cU}(M)$ is a doubly-graded vector space, with a `blob degree' and an `internal degree'. 

We'll write $\cT_k$ for the set of finite rooted trees with $k$ vertices labelled $1, \cdots, k$, and $\cT$ for the union of all $\cT_k$. We'll think of such a rooted tree as a category, with objects $1, \cdots, k$, and each morphism set either empty or a singleton, with $v \to w$ if $w$ is closer to a root of the tree than $v$. For such a tree $t \in \cT_k$, define $\abs{t} = k$, and for a vertex $v$ labelled by $i$ define $\sigma(v \in t) = (-1)^k$. We'll write $\hat{v}$ for the `parent' of a vertex $v$ if $v$ is not a root (that is, $\hat{v}$ is the unique vertex such that $v \to \hat{v}$ but there is no $w$ with $v \to w \to \hat{v}$. If $v$ is a root, we'll write $\hat{v}=\star$.

Given $v \in t$ labelled by $i$, there's a functor $\partial_v$ which removes the vertex $v$ and relabels all vertices labelled by $j>i$ with $j-1$. Notice that $$\sigma(v \in t) \sigma(w \in \partial_v t) = - \sigma(w \in t) \sigma(v \in \partial_w t).$$

Let $\operatorname{balls}(M)$ denote the category of open balls in $M$ with inclusions.

Given a tree $t \in \cT$ and a functor $b : t \to \operatorname{balls}(M)$ (one might call such a functor a `dendroidal ball' if one where so inclined), define  

\begin{equation*}

\cF(t,b) = \cF\left(M \setminus b(t)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ not a leaf}}} \cF\left(b(v) \setminus b(v' \to v)\right)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ a leaf}}} \cU\left(b(v)\right)\right)

\end{equation*}

and then

\begin{equation*}

\cB_*^{\cF,\cU}(M) = \DirectSum_{t \in \cT} \DirectSum_{\substack{\text{functors} \\ f: t \to \operatorname{balls}(M)}} \cF(t,f)

\end{equation*}

It becomes a chain complex by taking the homological degree to be the sum of the blob and internal degrees, and defining $d$ on $\cF(t,b)$ by

\begin{equation*}

d f = \sum_{v \in t} \partial_v f + \sum_{v' \in t \cup \{\star\}} d_{v'} f,

\end{equation*}

where if $f \in \cF(t,b)$ is an elementary tensor of the form $f = f_\star \tensor \Tensor_{v \in t} f_v$ with

\begin{align*}

f_\star & \in \cF(M \setminus b(t)) & \\

f_v       & \in \cF(b(v) \setminus b(v' \to v)) & \text{if $v$ is not a leaf} \\

f_v       & \in \cU(b(v)) & \text{if $v$ is a leaf}

\end{align*}

the terms $\partial_v f$ are again elementary tensors defined by

\begin{equation*}

(\partial_v f)_{v'} = \begin{cases} \sigma(v \in t) f_{\hat{v}} \circ f_v & \text{if $v' = \hat{v}$} \\ f_{v'} & \text{otherwise} \end{cases}

\end{equation*}

and the terms $d_v f$ are also elementary tensors defined by

\begin{equation*}

(d_v f)_{v'} = \begin{cases} (-1)^{\sum_{v \to v'} \deg f(v')} & \text{if $v'=v$} \\ f_v & \text{otherwise.} \end{cases}

\end{equation*}

\subsection{Product and gluing theorems}