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 \subsection{Definition of $n$-categories}
 \label{ss:n-cat-def}
 Before proceeding, we need more appropriate definitions of $n$-categories,
-$A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
+$A_\infty$ $n$-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
 a ``weak" $n$-category with ``strong duality".)
 The definitions presented below tie the categories more closely to the topology
 and avoid combinatorial questions about, for example, the minimal sufficient
 (e.g.\ categories whose morphisms are maps into spaces or decorated balls),
 it is easy to show that they
 satisfy our axioms.
 For examples of a more purely algebraic origin, one would typically need the combinatorial
 results that we have avoided here.
+\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
 There are many existing definitions of $n$-categories, with various intended uses.
 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$.
 the category of $k$-balls and
 homeomorphisms to the category of sets and bijections.
 \end{axiom}
-(Note: We usually omit the subscript $k$.)
+(Note: We often omit the subscript $k$.)
 We are being deliberately vague about what flavor of $k$-balls
 we are considering.
 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
 They could be topological or PL or smooth.
 (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 $n$-category.
 For simplicity, we will concentrate on the case of PL unoriented manifolds.
-(The ambitious reader may want to keep in mind two other classes of balls.
+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 to describe the blob complex of a fiber bundle with
+This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with
 base space $Y$.
 The second is balls equipped with a section of the tangent bundle, or the frame
-bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle.
+bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle.
 These can be used to define categories with less than the ``strong" duality we assume here,
-though we will not develop that idea fully in this paper.)
+though we will not develop that idea fully in this paper.
 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries
 of morphisms).
 The 0-sphere is unusual among spheres in that it is disconnected.
 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
 (Actually, this is only true in the oriented case, with 1-morphisms parameterized
 by {\it oriented} 1-balls.)
 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense.
 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc.
 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary.
-We prefer to not make the distinction in the first place.
+We prefer not to make the distinction in the first place.
 Instead, we will combine the domain and range into a single entity which we call the
 boundary of a morphism.
 Morphisms are modeled on balls, so their boundaries are modeled on spheres.
 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
 These maps, for various $X$, comprise a natural transformation of functors.
 \end{axiom}
-(Note that the first ``$\bd$" above is part of the data for the category,
+Note that the first ``$\bd$" above is part of the data for the category,
-while the second is the ordinary boundary of manifolds.)
+while the second is the ordinary boundary of manifolds.
 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
 Most of the examples of $n$-categories we are interested in are enriched in the following sense.
 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category
 (e.g.\ vector spaces, or modules over some ring, or chain complexes),
 \nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?}
 and all the structure maps of the $n$-category should be compatible with the auxiliary
 category structure.
-Note that this auxiliary structure is only in dimension $n$;
+Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then
-$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
+$\cC(Y; c)$ is just a plain set.
 \medskip
-(In order to simplify the exposition we have concentrated on the case of
+In order to simplify the exposition we have concentrated on the case of
 unoriented PL manifolds and avoided the question of what exactly we mean by
-the boundary a manifold with extra structure, such as an oriented manifold.
+the boundary of a manifold with extra structure, such as an oriented manifold.
 In general, all manifolds of dimension less than $n$ should be equipped with the germ
 of a thickening to dimension $n$, and this germ should carry whatever structure we have
 on $n$-manifolds.
 In addition, lower dimensional manifolds should be equipped with a framing
 of their normal bundle in the thickening; the framing keeps track of which
 side (iterated) bounded manifolds lie on.
 For example, the boundary of an oriented $n$-ball
 should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent
 bundle and a choice of direction in this bundle indicating
-which side the $n$-ball lies on.)
+which side the $n$-ball lies on.
 \medskip
 We have just argued that the boundary of a morphism has no preferred splitting into
 domain and range, but the converse meets with our approval.
 \end{tikzpicture}
 $$
 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
 Note that we insist on injectivity above.
-The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}.
+The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. \nn{we might want a more official looking proof...}
 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$.
 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$
 as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$.
 We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$
 a {\it restriction} map and write $\res_{B_i}(a)$
 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$.
-More generally, we also include under the rubric ``restriction map" the
+More generally, we also include under the rubric ``restriction map"
 the boundary maps of Axiom \ref{nca-boundary} above,
 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition
 of restriction maps.
 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$
 ($i = 1, 2$, notation from previous paragraph).
 $$
 \caption{From two balls to one ball.}\label{blah5}\end{figure}
 \begin{axiom}[Strict associativity] \label{nca-assoc}
 The composition (gluing) maps above are strictly associative.
-Given any splitting of a ball $B$ into smaller balls $B_1,\ldots,B_m$,
+Given any splitting of a ball $B$ into smaller balls
-any sequence of gluings of the smaller balls yields the same result.
+$$\bigsqcup B_i \to B,$$
+any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result.
 \end{axiom}
 \begin{figure}[!ht]
 $$\mathfig{.65}{ncat/strict-associativity}$$
 \caption{An example of strict associativity.}\label{blah6}\end{figure}
-We'll use the notations  $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$.
+We'll use the notation  $a\bullet b$ for the glued together field $\gl_Y(a, b)$.
 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$
 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.
 %Compositions of boundary and restriction maps will also be called restriction maps.
 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a
 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$.
 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$.
 We will call elements of $\cC(B)_Y$ morphisms which are
 ``splittable along $Y$'' or ``transverse to $Y$''.
 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.
-More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls.
+More generally, let $\alpha$ be a splitting of $X$ into smaller balls.
 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from
 the smaller balls to $X$.
 We  say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$".
-In situations where the subdivision is notationally anonymous, we will write
+In situations where the splitting is notationally anonymous, we will write
 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)
-the unnamed subdivision.
+the unnamed splitting.
-If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
+If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous
-subdivision of $\bd X$ and no competing subdivision of $X$.
+decomposition of $\bd X$ and no competing splitting of $X$.
 The above two composition axioms are equivalent to the following one,
 which we state in slightly vague form.
 \xxpar{Multi-composition:}
-{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball
+{Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball
 into small $k$-balls, there is a
 map from an appropriate subset (like a fibered product)
 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$,
 and these various $m$-fold composition maps satisfy an
 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).}
 	d: \Delta^{k+m}\to\Delta^k .
 \]
 (We thank Kevin Costello for suggesting this approach.)
 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,
-and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
+and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
 $l \le m$, with $l$ depending on $x$.
 It is easy to see that a composition of pinched products is again a pinched product.
 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction
 $\pi:E'\to \pi(E')$ is again a pinched product.
 A {union} of pinched products is a decomposition $E = \cup_i E_i$
 such that each $E_i\sub E$ is a sub pinched product.
 (See Figure \ref{pinched_prod_unions}.)
 We start with the plain $n$-category case.
 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
-Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.
+Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$.
 \end{axiom}
 This axiom needs to be strengthened to force product morphisms to act as the identity.
 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
 Let $J$ be a 1-ball (interval).
 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
-(Here we use the ``pinched" version of $Y\times J$.
+(Here we use $Y\times J$ with boundary entirely pinched.)
-\nn{do we need notation for this?})
 We define a map
 \begin{eqnarray*}
 	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
 	a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .
 \end{eqnarray*}
 a diagram like the one in Theorem \ref{thm:CH} commutes.
 %\nn{repeat diagram here?}
 %\nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}
 \end{axiom}
-We should strengthen the above axiom to apply to families of collar maps.
+We should strengthen the above $A_\infty$ axiom to apply to families of collar maps.
 To do this we need to explain how collar maps form a topological space.
 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
 and we can replace the class of all intervals $J$ with intervals contained in $\r$.
 Having chains on the space of collar maps act gives rise to coherence maps involving
 weak identities.
 (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 $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ 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,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases}
+\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
 a colimit construction; see \S \ref{ss:ncat_fields} below.
 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$,
 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 the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
+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 $\cE$.
 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
 are splittable along this decomposition.
 \begin{defn}
 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
-For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
+For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
 \begin{equation}
 \label{eq:psi-C}
 	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
 \end{equation}
 where the restrictions to the various pieces of shared boundaries amongst the cells
 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$.
 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in
 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate
 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect).
-Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
+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 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
 We now give more concrete descriptions of the above colimits.
 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set,
 the colimit is
 \[
-	\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim ,
+	\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim ,
 \]
 where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation
 induced by refinement and gluing.
 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold,
 we can take
 \begin{equation*}
-	\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K
+	\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,
 \end{equation*}
 where $K$ is the vector space spanned by elements $a - g(a)$, with
 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)
 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.
 Such sequences (for all $m$) form a simplicial set in $\cell(W)$.
 Define $\cl{\cC}(W)$ as a vector space via
 \[
 	\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
 \]
-where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
+where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$.
 Elements of a summand indexed by an $m$-sequence will be call $m$-simplices.
 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
 summands plus another term using the differential of the simplicial set of $m$-sequences.
 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define
 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 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.
-We state the final axiom, on actions of homeomorphisms, differently in the two cases.
+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
 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
 We call $B$ the ball and $N$ the marking.
 A homeomorphism between marked $k$-balls is a homeomorphism of balls which
 (The union is along $N\times \bd W$.)
 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be
 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.)
 \begin{figure}[!ht]
-$$\mathfig{.8}{ncat/boundary-collar}$$
+$$\mathfig{.55}{ncat/boundary-collar}$$
 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure}
 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
 Call such a thing a {marked $k{-}1$-hemisphere}.
 \end{figure}
 The above three axioms are equivalent to the following axiom,
 which we state in slightly vague form.
-\nn{need figure for this}
 \xxpar{Module multi-composition:}
-{Given any decomposition
+{Given any splitting
 \[
-	M =  X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q
+	X_1 \sqcup\cdots\sqcup X_p \sqcup M_1\sqcup\cdots\sqcup M_q \to M
 \]
 of a marked $k$-ball $M$
 into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a
 map from an appropriate subset (like a fibered product)
 of
 \end{example}
 \begin{example}[Examples from the blob complex] \label{bc-module-example}
 \rm
 In the previous example, we can instead define
-$\cF(Y)(M)\deq \bc_*^\cF((B\times W) \cup (N\times Y); c)$ (when $\dim(M) = n$)
+$\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$)
 and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in
 Example \ref{ex:blob-complexes-of-balls}.
 \end{example}
 Fix a topological $n$-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 similar to
+We will define a set $\cC(W, \cN)$ using a colimit construction very similar to
 the one appearing in \S \ref{ss:ncat_fields} above.
 (If $k = n$ and our $n$-categories are enriched, then
 $\cC(W, \cN)$ will have additional structure; see below.)
-Define a permissible decomposition of $W$ to be a decomposition
+Define a permissible decomposition of $W$ to be a map
 \[
-	W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) ,
+	\left(\bigsqcup_a X_a\right) \sqcup \left(\bigsqcup_{i,b} M_{ib}\right)  \to W,
 \]
-where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and
+where each $X_a$ is a plain $k$-ball disjoint, in $W$, from $\cup Y_i$, and
-each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$,
+each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$  (once mapped into $W$),
-with $M_{ib}\cap Y_i$ being the marking.
+with $M_{ib}\cap Y_i$ being the marking, which extends to a ball decomposition in the sense of Definition \ref{defn:gluing-decomposition}.
 (See Figure \ref{mblabel}.)
 \begin{figure}[t]
 \begin{equation*}
 \mathfig{.4}{ncat/mblabel}
 \end{equation*}

changeset 494	cb76847c439e
parent 479	cfad13b6b1e5
child 497	18b742b1b308