5 [some things to cover in the intro]

     5 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:

     6 \begin{itemize}

     6 \begin{itemize}

     7 \item explain relation between old and new blob complex definitions

     7 \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}.)

     8 \item overview of sections

     8 \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}.)

     9 \item state main properties of blob complex (already mostly done below)

     9 \item When $\cC = k[t]$, thought of as an n-category, we have $$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ (See \S \ref{sec:comm_alg}.)

    10 \item give multiple motivations/viewpoints for blob complex: (1) derived cat

    10 \end{itemize}

    11 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}).

    12 

    13 The blob complex definition is motivated by \nn{ continue here ...} give multiple motivations/viewpoints for blob complex: (1) derived cat

    15 \item hope to apply to Kh, contact, (other examples?) in the future

    18 

    19 We expect applications of the blob complex to \nn{ ... } but do not address these in this paper.

    20 \nn{hope to apply to Kh, contact, (other examples?) in the future}

    21 

    22 

    23 \subsubsection{Structure of the paper}

    24 

    25 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.

    26 

    27 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. When $n=1$ these reduce to the usual $A_\infty$ categories.

    28 

    29 In the third part of the paper (section \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.

    30 

    31 

    32 [some things to cover in the intro]

    33 \begin{itemize}

    34 \item explain relation between old and new blob complex definitions

    35 \item overview of sections

   152 

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

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

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

   195 Further, for

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

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

   198 \begin{equation*}

   199 \xymatrix@C+2cm{

   200      \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) \\

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

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

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

   204 }

   205 \end{equation*}

   177 instead of a garden variety $n$-category.

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