--- a/text/a_inf_blob.tex	Fri Jan 07 14:19:50 2011 -0800
+++ b/text/a_inf_blob.tex	Fri Jan 07 14:40:58 2011 -0800
@@ -208,7 +208,7 @@
 (See Example \ref{ex:blob-complexes-of-balls}.)
 Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
 These two categories are equivalent, but since we do not define functors between
-topological $n$-categories in this paper we are unable to say precisely
+disk-like $n$-categories in this paper we are unable to say precisely
 what ``equivalent" means in this context.
 We hope to include this stronger result in a future paper.
 

--- a/text/appendixes/comparing_defs.tex	Fri Jan 07 14:19:50 2011 -0800
+++ b/text/appendixes/comparing_defs.tex	Fri Jan 07 14:40:58 2011 -0800
@@ -7,13 +7,13 @@
 a topological $n$-category from a traditional $n$-category; the morphisms of the 
 topological $n$-category are string diagrams labeled by the traditional $n$-category.
 In this appendix we sketch how to go the other direction, for $n=1$ and 2.
-The basic recipe, given a topological $n$-category $\cC$, is to define the $k$-morphisms
+The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms
 of the corresponding traditional $n$-category to be $\cC(B^k)$, where
 $B^k$ is the {\it standard} $k$-ball.
 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms.
-One should also show that composing the two arrows (between traditional and topological $n$-categories)
+One should also show that composing the two arrows (between traditional and disk-like $n$-categories)
 yields the appropriate sort of equivalence on each side.
-Since we haven't given a definition for functors between topological $n$-categories
+Since we haven't given a definition for functors between disk-like $n$-categories
 (the paper is already too long!), we do not pursue this here.
 
 We emphasize that we are just sketching some of the main ideas in this appendix ---
@@ -24,7 +24,7 @@
 
 \subsection{1-categories over \texorpdfstring{$\Set$ or $\Vect$}{Set or Vect}}
 \label{ssec:1-cats}
-Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$.
+Given a disk-like $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$.
 This construction is quite straightforward, but we include the details for the sake of completeness, 
 because it illustrates the role of structures (e.g. orientations, spin structures, etc) 
 on the underlying manifolds, and 
@@ -70,7 +70,7 @@
 \medskip
 
 In the other direction, given a $1$-category $C$
-(with objects $C^0$ and morphisms $C^1$) we will construct a topological
+(with objects $C^0$ and morphisms $C^1$) we will construct a disk-like
 $1$-category $t(C)$.
 
 If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$.
@@ -79,7 +79,7 @@
 Homeomorphisms isotopic to the identity act trivially.
 If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure
 to define the action of homeomorphisms not isotopic to the identity
-(and get, e.g., an unoriented topological 1-category).
+(and get, e.g., an unoriented disk-like 1-category).
 
 The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$.
 
@@ -100,13 +100,13 @@
 
 \medskip
 
-Similar arguments show that modules for topological 1-categories are essentially
+Similar arguments show that modules for disk-like 1-categories are essentially
 the same thing as traditional modules for traditional 1-categories.
 
 
 \subsection{Pivotal 2-categories}
 \label{ssec:2-cats}
-Let $\cC$ be a topological 2-category.
+Let $\cC$ be a disk-like 2-category.
 We will construct from $\cC$ a traditional pivotal 2-category.
 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
 
@@ -559,11 +559,11 @@
 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories}
 \label{sec:comparing-A-infty}
 In this section, we make contact between the usual definition of an $A_\infty$ category 
-and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}.
+and our definition of a disk-like $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}.
 
 \medskip
 
-Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" 
+Given a disk-like $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" 
 $A_\infty$ $1$-category $A$ as follows.
 The objects of $A$ are $\cC(pt)$.
 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$
@@ -605,7 +605,7 @@
 Operad associativity for $A$ implies that this gluing map is independent of the choice of
 $g$ and the choice of representative $(f_i, a_i)$.
 
-It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category.
+It is straightforward to verify the remaining axioms for a disk-like $A_\infty$ 1-category.
 
 
 

--- a/text/intro.tex	Fri Jan 07 14:19:50 2011 -0800
+++ b/text/intro.tex	Fri Jan 07 14:40:58 2011 -0800
@@ -63,29 +63,30 @@
 Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another 
 definition of an $n$-category, or rather a definition of an $n$-category with strong duality.
 (Removing the duality conditions from our definition would make it more complicated rather than less.) 
-We call these ``topological $n$-categories'', to differentiate them from previous versions.
+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 topological $n$-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 
 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 
-topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
+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.
 
-In \S \ref{ss:ncat_fields}  we explain how to construct a system of fields from a topological $n$-category 
+In \S \ref{ss:ncat_fields}  we explain how to construct a system of fields from a disk-like $n$-category 
 (using a colimit along certain decompositions of a manifold into balls). 
 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ 
 with the system of fields constructed from the $n$-category $\cC$. 
 %\nn{KW: I don't think we use this notational convention any more, right?}
 In \S \ref{sec:ainfblob} we give an alternative definition 
 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit).
-Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an 
+Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-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 Theorem \ref{thm:gluing} below.
@@ -105,7 +106,7 @@
 \newcommand{\yyc}{6}
 
 \node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category};
-\node[box] at (\xxa,\yya) (C) {$\cC$ \\ a topological \\ $n$-category};
+\node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category};
 \node[box] at (\xxb,\yya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space};
 \node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations};
 \node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex};
@@ -148,7 +149,7 @@
 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 topological $n$-category, in terms of the topology of $M$.
+%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)
@@ -371,15 +372,15 @@
 for any homeomorphic pair $X$ and $Y$, 
 satisfying corresponding conditions.
 
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields.
-Below, when we talk about the blob complex for a topological $n$-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 topological $n$-categories:
+In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields.
+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}
-Let $\cC$ be  a topological $n$-category.
+Let $\cC$ be  an ordinary $n$-category.
 Let $Y$ be an $n{-}k$-manifold. 
 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, 
 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set 
@@ -389,12 +390,12 @@
 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 a topological $n$-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 a topological $n$-category; this is described in \S \ref{sec:ainfblob}.
+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 notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.
 
@@ -412,9 +413,9 @@
 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
 (see \S \ref{ss:product-formula}).
 
-Fix a topological $n$-category $\cC$, which we'll omit from the notation.
+Fix a disk-like $n$-category $\cC$, which we'll omit from the notation.
 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
-(See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
+(See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
 
 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}}
 

--- a/text/ncat.tex	Fri Jan 07 14:19:50 2011 -0800
+++ b/text/ncat.tex	Fri Jan 07 14:40:58 2011 -0800
@@ -23,6 +23,8 @@
 For examples of a more purely algebraic origin, one would typically need the combinatorial
 results that we have avoided here.
 
+See \S\ref{n-cat-names} for a discussion of $n$-category terminology.
+
 %\nn{Say something explicit about Lurie's work here? 
 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen}
 
@@ -378,7 +380,7 @@
 \caption{Examples of pinched products}\label{pinched_prods}
 \end{figure}
 (The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs}
-where we construct a traditional category from a topological category.)
+where we construct a traditional category from a disk-like category.)
 Define a {\it pinched product} to be a map
 \[
 	\pi: E\to X
@@ -668,7 +670,7 @@
 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases}
 \end{align*}
 This $n$-category can be thought of as the local part of the fields.
-Conversely, given a topological $n$-category we can construct a system of fields via 
+Conversely, given a disk-like $n$-category we can construct a system of fields via 
 a colimit construction; see \S \ref{ss:ncat_fields} below.
 
 In the $n$-category axioms above we have intermingled data and properties for expository reasons.
@@ -855,16 +857,16 @@
 \end{example}
 
 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
-Notice that with $F$ a point, the above example is a construction turning a topological 
+Notice that with $F$ a point, the above example is a construction turning an ordinary 
 $n$-category $\cC$ into an $A_\infty$ $n$-category.
 We think of this as providing a ``free resolution" 
-of the topological $n$-category. 
+of the ordinary $n$-category. 
 %\nn{say something about cofibrant replacements?}
 In fact, there is also a trivial, but mostly uninteresting, way to do this: 
 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, 
 and take $\CD{B}$ to act trivially. 
 
-Beware that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ 
+Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ 
 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
 It's easy to see that with $n=0$, the corresponding system of fields is just 
 linear combinations of connected components of $T$, and the local relations are trivial.
@@ -927,7 +929,7 @@
 also comes from the $\cE\cB_n$ action on $A$.
 %\nn{should we spell this out?}
 
-Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to 
 an $\cE\cB_n$-algebra.
 %\nn{The paper is already long; is it worth giving details here?}
@@ -1195,7 +1197,7 @@
 This will be explained in more detail as we present the axioms.
 
 Throughout, we fix an $n$-category $\cC$.
-For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category.
+For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category.
 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases.
 
 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
@@ -1509,7 +1511,7 @@
 
 \medskip
 
-We now give some examples of modules over topological and $A_\infty$ $n$-categories.
+We now give some examples of modules over ordinary and $A_\infty$ $n$-categories.
 
 \begin{example}[Examples from TQFTs]
 \rm
@@ -1552,7 +1554,7 @@
 \subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}
 
-Fix a topological $n$-category or $A_\infty$ $n$-category  $\cC$.
+Fix an ordinary $n$-category or $A_\infty$ $n$-category  $\cC$.
 Let $W$ be a $k$-manifold ($k\le n$),
 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
@@ -1720,7 +1722,7 @@
 
 %The results of this subsection are not needed for the rest of the paper,
 %so we will skimp on details in a couple of places. We have included this mostly 
-%for the sake of comparing our notion of a topological $n$-category to other definitions.
+%for the sake of comparing our notion of a disk-like $n$-category to other definitions.
 
 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces.