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 We call these ``disk-like $n$-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.
 (See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.)
 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 disk-like $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball.
+We try to be as lax as possible: a disk-like $n$-category associates a
+vector space to every $B$ homeomorphic to the $n$-ball.
 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.
-For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of
+For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to
+each such $B$ and ask that the action of
 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms.
-The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a
+The axioms for an $A_\infty$ $n$-category are designed to capture two main examples:
+the blob complexes of $n$-balls labelled by a
 disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category
 of sphere modules.
 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors.
 The appendices prove technical results about $\CH{M}$ and
 make connections between our definitions of $n$-categories and familiar 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.
 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra,
 %thought of as a disk-like $n$-category, in terms of the topology of $M$.
-%%%% this is said later in the intro
-%Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes)
-%even when we could work in greater generality (symmetric monoidal categories, model categories, etc.).
-%\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
 \subsection{Motivations}
 \label{sec:motivations}
 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary,
 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ 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 $X \to X_\mathrm{gl}$, there is
 a natural map
 \[
 	\bc_*(X) \to \bc_*(X_\mathrm{gl})
 \]
 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps
 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
 for any homeomorphic pair $X$ and $Y$,
 satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields.
+In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories,
-Below, when we talk about the blob complex for a disk-like $n$-category, we are implicitly passing first to this associated system of fields.
+from which we can construct systems of fields.
-Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. In that section we describe how to use the blob complex to construct $A_\infty$ $n$-categories from ordinary $n$-categories:
+Below, when we talk about the blob complex for a disk-like $n$-category,
+we are implicitly passing first to this associated system of fields.
+Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
+In that section we describe how to use the blob complex to
+construct $A_\infty$ $n$-categories from ordinary $n$-categories:
 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}}
 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category]
 %\label{thm:blobs-ainfty}
 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.)
 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in
 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.
 \end{ex:blob-complexes-of-balls}
 \begin{rem}
-Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category.
+Perhaps the most interesting case is when $Y$ is just a point;
+then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category.
 We think of this $A_\infty$ $n$-category as a free resolution.
 \end{rem}
 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}.
-The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example.
+The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
+The next theorem describes the blob complex for product manifolds,
+in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example.
 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.
 \newtheorem*{thm:product}{Theorem \ref{thm:product}}
 \begin{thm:product}[Product formula]
 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
 Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}).
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology
+(see Example \ref{ex:blob-complexes-of-balls}).
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
 \]
 \end{thm:product}
 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls.
 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied
 by ``disk-like".
 (But beware: disks can come in various flavors, and some of them, such as framed disks,
 don't actually imply much duality.)
-Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. It will thus be a theorem that our disk-like 2-categories are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}.
+Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense.
+It will thus be a theorem that our disk-like 2-categories
+are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}.
 Finally, we need a general name for isomorphisms between balls, where the balls could be
 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof.
 We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should
 keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on)
 Alan Wilder,
 Dmitri Pavlov,
 Ansgar Schneider,
 and
 Dan Berwick-Evans.
-\nn{need to double-check this list once the reading course is over}
+During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at
-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. We'd like to thank the Aspen Center for Physics for the pleasant and productive
+Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.
+We'd like to thank the Aspen Center for Physics for the pleasant and productive
 environment provided there during the final preparation of this manuscript.

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