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 %!TEX root = ../blob1.tex
 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
-\section{Disk-like \texorpdfstring{$n$}{n}-categories and their modules}
+\section{\texorpdfstring{$n$}{n}-categories and their modules}
 \label{sec:ncats}
-\subsection{Definition of disk-like \texorpdfstring{$n$}{n}-categories}
+\subsection{Definition of \texorpdfstring{$n$}{n}-categories}
 \label{ss:n-cat-def}
 Before proceeding, we need more appropriate definitions of $n$-categories,
 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules.
 (As is the case throughout this paper, by ``$n$-category" we mean some notion of
 %\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}
 \medskip
-The axioms for a disk-like $n$-category are spread throughout this section.
+The axioms for an $n$-category are spread throughout this section.
-Collecting these together, a disk-like $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
+Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
 \ref{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and  \ref{axiom:vcones}.
-For an enriched disk-like $n$-category we add Axiom \ref{axiom:enriched}.
+For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
-For an $A_\infty$ disk-like $n$-category, we replace
+For an $A_\infty$ $n$-category, we replace
 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms
 for $k{-}1$-morphisms.
 Readers who prefer things to be presented in a strictly logical order should read this
 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
 They could be topological or PL or smooth.
 %\nn{need to check whether this makes much difference}
 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
 to be fussier about corners and boundaries.)
-For each flavor of manifold there is a corresponding flavor of disk-like $n$-category.
+For each flavor of manifold there is a corresponding flavor of $n$-category.
 For simplicity, we will concentrate on the case of PL unoriented manifolds.
 An ambitious reader may want to keep in mind two other classes of balls.
 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}).
 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with
 decomposition which has common refinements with each of the original two decompositions.
 \medskip
-This completes the definition of a disk-like $n$-category.
+This completes the definition of an $n$-category.
-Next we define enriched disk-like $n$-categories.
+Next we define enriched $n$-categories.
 \medskip
 Most of the examples of $n$-categories we are interested in are enriched in the following sense.
 \item topological spaces with product and disjoint union.
 \end{itemize}
 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure.
 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.)
-Before stating the revised axioms for a disk-like $n$-category enriched in a distributive monoidal category,
+Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
 we need a preliminary definition.
 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the
 category $\bbc$ of {\it $n$-balls with boundary conditions}.
 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition".
 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are
 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
 %Let $\pi_0(\bbc)$ denote
-\begin{axiom}[Enriched disk-like $n$-categories]
+\begin{axiom}[Enriched $n$-categories]
 \label{axiom:enriched}
 Let $\cS$ be a distributive symmetric monoidal category.
-A disk-like $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
+An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
 and modifies the axioms for $k=n$ as follows:
 \begin{itemize}
 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$.
 %[already said this above.  ack]  Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$.
 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially
 When the enriching category $\cS$ is chain complexes or topological spaces,
 or more generally an appropriate sort of $\infty$-category,
 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
 to require that families of homeomorphisms act
-and obtain what we shall call an $A_\infty$ disk-like $n$-category.
+and obtain what we shall call an $A_\infty$ $n$-category.
 \noop{
 We believe that abstract definitions should be guided by diverse collections
 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories
 makes us reluctant to commit to an all-encompassing general definition.
 In fact, compatibility implies less than this.
 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor.
 (This is the example most relevant to this paper.)
 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action
 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
-And indeed, this is true for our main example of an $A_\infty$ disk-like $n$-category based on the blob construction.
+And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction.
 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions,
 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom.
 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families}
 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a
 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom.
 %since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.
 For future reference we make the following definition.
 \begin{defn}
-A {\em strict $A_\infty$ disk-like $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
+A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
 \end{defn}
 \noop{
 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
 into a ordinary $n$-category (enriched over graded groups).
 }
 \medskip
-We define a $j$ times monoidal disk-like $n$-category to be a disk-like $(n{+}j)$-category $\cC$ where
+We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where
 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$.
 See Example \ref{ex:bord-cat}.
 \medskip
-The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category
+The alert reader will have already noticed that our definition of an (ordinary) $n$-category
 is extremely similar to our definition of a system of fields.
 There are two differences.
 First, for the $n$-category definition we restrict our attention to balls
 (and their boundaries), while for fields we consider all manifolds.
 Second,  in category definition we directly impose isotopy
 invariance in dimension $n$, while in the fields definition we
 instead remember a subspace of local relations which contain differences of isotopic fields.
 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
-Thus a system of fields and local relations $(\cF,U)$ determines a disk-like $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
+Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
 balls and, at level $n$, quotienting out by the local relations:
 \begin{align*}
 \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.
 \medskip
 In the $n$-category axioms above we have intermingled data and properties for expository reasons.
 Here's a summary of the definition which segregates the data from the properties.
-A disk-like $n$-category consists of the following data:
+An $n$-category consists of the following data:
 \begin{itemize}
 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition});
 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product});
 \item For ordinary categories, invariance of $n$-morphisms under extended isotopies
 and collar maps (Axiom \ref{axiom:extended-isotopies}).
 \end{itemize}
-\subsection{Examples of disk-like \texorpdfstring{$n$}{n}-categories}
+\subsection{Examples of \texorpdfstring{$n$}{n}-categories}
 \label{ss:ncat-examples}
 We now describe several classes of examples of $n$-categories satisfying our axioms.
 We typically specify only the morphisms; the rest of the data for the category
 	C_*(\Maps_c(X \to T)),
 \]
 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
 and $C_*$ denotes singular chains.
 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$,
-we get an $A_\infty$ disk-like $n$-category enriched over spaces.
+we get an $A_\infty$ $n$-category enriched over spaces.
 \end{example}
 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
 \begin{example}[Blob complexes of balls (with a fiber)]
 \rm
 \label{ex:blob-complexes-of-balls}
 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
-We will define an $A_\infty$ disk-like $k$-category $\cC$.
+We will define an $A_\infty$ $k$-category $\cC$.
 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
 When $X$ is an $k$-ball,
 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
 where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
 \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 an ordinary disk-like
+Notice that with $F$ a point, the above example is a construction turning an ordinary
-$n$-category $\cC$ into an $A_\infty$ disk-like $n$-category.
+$n$-category $\cC$ into an $A_\infty$ $n$-category.
 We think of this as providing a ``free resolution"
-of the ordinary disk-like $n$-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 ordinary disk-like $n$-category $\pi_{\leq n}(T)$
+Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$
-is not the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$.
+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.
 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version]
 \rm
 \label{ex:e-n-alg}
 Let $A$ be an $\cE\cB_n$-algebra.
 Note that this implies a $\Diff(B^n)$ action on $A$,
 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
-We will define a strict $A_\infty$ disk-like $n$-category $\cC^A$.
+We will define a strict $A_\infty$ $n$-category $\cC^A$.
 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
 In other words, the $k$-morphisms are trivial for $k<n$.
 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
 (Plain colimit, not homotopy colimit.)
 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of
 embedded balls into a single larger embedded ball.
 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and
 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.
 Alternatively and more simply, we could define $\cC^A(X)$ to be
 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.
-The remaining data for the $A_\infty$ disk-like $n$-category
+The remaining data for the $A_\infty$ $n$-category
 --- composition and $\Diff(X\to X')$ action ---
 also comes from the $\cE\cB_n$ action on $A$.
 %\nn{should we spell this out?}
-Conversely, one can show that a strict $A_\infty$  disk-like $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like strict $A_\infty$ $n$-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?}
 % According to the referee, yes it is...
 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball.
 \end{example}
 \subsection{From balls to manifolds}
 \label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we show how to extend a disk-like $n$-category $\cC$ as described above
+In this section we show how to extend an $n$-category $\cC$ as described above
 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
-In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a
+In the case of ordinary $n$-categories, this construction factors into a construction of a
 system of fields and local relations, followed by the usual TQFT definition of a
 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
-For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
+For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
-Recall that we can take a ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution",
+Recall that we can take a ordinary $n$-category $\cC$ and pass to the ``free resolution",
-an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
+an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
 (recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant
-for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the
+for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the
 same as the original blob complex for $M$ with coefficients in $\cC$.
 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def},
 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls,
 so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
 \medskip
 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
-A disk-like $n$-category $\cC$ provides a functor from this poset to the category of sets,
+An $n$-category $\cC$ provides a functor from this poset to the category of sets,
 and we  will define $\cl{\cC}(W)$ as a suitable colimit
 (or homotopy colimit in the $A_\infty$ case) of this functor.
 We'll later give a more explicit description of this colimit.
-In the case that the disk-like $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain
 complexes to $n$-balls with boundary data),
 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into
 subsets according to boundary data, and each of these subsets has the appropriate structure
 (e.g. a vector space or chain complex).
 \end{equation*}
 \caption{A small part of $\cell(W)$}
 \label{partofJfig}
 \end{figure}
-A disk-like $n$-category $\cC$ determines
+An $n$-category $\cC$ determines
 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets
 (possibly with additional structure if $k=n$).
 Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$).
 We will define $\psi_{\cC;W}(x)$ to be a certain subset of $\prod_a \cC(X_a)$.
 Roughly speaking, $\psi_{\cC;W}(x)$ is the subset where the restriction maps from
 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$:
 \begin{defn}[System of fields functor]
 \label{def:colim-fields}
-If $\cC$ is a disk-like $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
+If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
 That is, for each decomposition $x$ there is a map
 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
 above, and $\cl{\cC}(W)$ is universal with respect to these properties.
 \end{defn}
 \begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ disk-like $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
+When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
 is defined as above, as the colimit of $\psi_{\cC;W}$.
 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
 \end{defn}
 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$
 %\nn{need to finish explaining why we have a system of fields;
 %define $k$-cat $\cC(\cdot\times W)$}
 \subsection{Modules}
-Next we define ordinary and $A_\infty$ disk-like $n$-category modules.
+Next we define ordinary and $A_\infty$ $n$-category modules.
-The definition will be very similar to that of disk-like $n$-categories,
+The definition will be very similar to that of $n$-categories,
 but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
 in the context of an $m{+}1$-dimensional TQFT.
-Such a $W$ gives rise to a module for the disk-like $n$-category associated to $\bd W$.
+Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
 This will be explained in more detail as we present the axioms.
-Throughout, we fix a disk-like $n$-category $\cC$.
+Throughout, we fix an $n$-category $\cC$.
 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
 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
 These maps, for various $M$, comprise a natural transformation of functors.}
 \end{module-axiom}
 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
-If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
+If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
 \begin{lem}[Boundary from domain and range]
 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$),
 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere.
 \]
 ($Y$ could be either a marked or plain ball.)
 \end{enumerate}
 \end{module-axiom}
-As in the disk-like $n$-category definition, once we have product morphisms we can define
+As in the $n$-category definition, once we have product morphisms we can define
 collar maps $\cM(M)\to \cM(M)$.
 Note that there are two cases:
 the collar could intersect the marking of the marked ball $M$, in which case
 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking,
 in which case we use a product on a morphism of $\cC$.
 $a$ along a map associated to $\pi$.
 \medskip
 There are two alternatives for the next axiom, according whether we are defining
-modules for ordinary or $A_\infty$ disk-like $n$-categories.
+modules for ordinary $n$-categories or $A_\infty$ $n$-categories.
 In the ordinary case we require
 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$]
 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts
 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity.
 As with the $n$-category version of the above axiom, we should also have families of collar maps act.
 \medskip
-Note that the above axioms imply that a disk-like $n$-category module has the structure
+Note that the above axioms imply that an $n$-category module has the structure
-of a disk-like $n{-}1$-category.
+of an $n{-}1$-category.
 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$,
 where $X$ is a $k$-ball and in the product $X\times J$ we pinch
 above the non-marked boundary component of $J$.
 (More specifically, we collapse $X\times P$ to a single point, where
 $P$ is the non-marked boundary component of $J$.)
-Then $\cE$ has the structure of a disk-like $n{-}1$-category.
+Then $\cE$ has the structure of an $n{-}1$-category.
 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds
 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$).
 In this case ($k=1$ and oriented or Spin), there are two types
 of marked 1-balls, call them left-marked and right-marked,
 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$),
 there is no left/right module distinction.
 \medskip
-We now give some examples of modules over ordinary and $A_\infty$ disk-like $n$-categories.
+We now give some examples of modules over ordinary and $A_\infty$ $n$-categories.
 \begin{example}[Examples from TQFTs]
 \rm
 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold,
-and $\cF(W)$ the disk-like $j$-category associated to $W$.
+and $\cF(W)$ the $j$-category associated to $W$.
 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$.
 Define a $\cF(W)$ module $\cF(Y)$ as follows.
 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let
 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$.
 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let
 \begin{example}[Examples from the blob complex] \label{bc-module-example}
 \rm
 In the previous example, we can instead define
 $\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$)
-and get a module for the $A_\infty$ disk-like $n$-category associated to $\cF$ as in
+and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in
 Example \ref{ex:blob-complexes-of-balls}.
 \end{example}
 \begin{example}
 \subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}
-Fix an ordinary or $A_\infty$ disk-like $n$-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 $Y_i$.
 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to
 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define
 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold
 $D\times Y_i \sub \bd(D\times W)$.
 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
-has the structure of a disk-like $n{-}k$-category.
+has the structure of an $n{-}k$-category.
 \medskip
 We will use a simple special case of the above
 construction to define tensor products
 of modules.
-Let $\cM_1$ and $\cM_2$ be modules for a disk-like $n$-category $\cC$.
+Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$.
 (If $k=1$ and our manifolds are oriented, then one should be
 a left module and the other a right module.)
 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
 Define the tensor product $\cM_1 \tensor \cM_2$ to be the
-disk-like $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
+$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
 This of course depends (functorially)
 on the choice of 1-ball $J$.
 We will define a more general self tensor product (categorified coend) below.
 then compose the module maps.
 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}.
 \medskip
-We end this subsection with some remarks about Morita equivalence of disk-like $n$-categories.
+We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent
 objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners.
-Similarly, we define two disk-like $n$-categories to be Morita equivalent if they are equivalent objects in the
+Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
 $n{+}1$-category of sphere modules.
-Because of the strong duality enjoyed by disk-like $n$-categories, the data for such an equivalence lives only in
+Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in
 dimensions 1 and $n+1$ (the middle dimensions come along for free).
 The $n{+}1$-dimensional part of the data must be invertible and satisfy
 identities corresponding to Morse cancellations in $n$-manifolds.
 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.
-Let $\cC$ and $\cD$ be (unoriented) disk-like 2-categories.
+Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories.
 Let $\cS$ denote the 3-category of 2-category sphere modules.
 The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$
 (categorified bimodule) connecting $\cC$ and $\cD$.
 Because of the full unoriented symmetry, this can also be thought of as a
 0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$.

changeset 888	a0fd6e620926
parent 865	7abe7642265e
child 889	70e947e15f57