--- a/text/ncat.tex Fri Jun 04 08:15:08 2010 -0700
+++ b/text/ncat.tex Fri Jun 04 11:42:07 2010 -0700
@@ -24,7 +24,9 @@
\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 $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical shape.
+There are many existing definitions of $n$-categories, with various intended uses.
+In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$.
+Generally, these sets are indexed by instances of a certain typical shape.
Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on).
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$,
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,
@@ -33,8 +35,10 @@
Still other definitions (see, for example, \cite{MR2094071})
model the $k$-morphisms on more complicated combinatorial polyhedra.
-For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic
-to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
+For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball.
+Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic
+to the standard $k$-ball.
+By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
We {\it do not} assume that it is equipped with a
preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
@@ -79,7 +83,10 @@
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 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.
+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.
Instead, we will combine the domain and range into a single entity which we call the
boundary of a morphism.
@@ -98,7 +105,9 @@
homeomorphisms to the category of sets and bijections.
\end{lem}
-We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels.
+We postpone the proof \todo{} of this result until after we've actually given all the axioms.
+Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$,
+along with the data described in the other Axioms at lower levels.
%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
@@ -284,7 +293,9 @@
The next axiom is related to identity morphisms, though that might not be immediately obvious.
\begin{axiom}[Product (identity) morphisms]
-For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions.
+For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$,
+usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
+These maps must satisfy the following conditions.
\begin{enumerate}
\item
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
@@ -478,7 +489,8 @@
(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 have do not expect isotopy invariance on fields
-but 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.)
+but 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
balls and, at level $n$, quotienting out by the local relations:
\begin{align*}
@@ -497,7 +509,8 @@
\begin{example}[Maps to a space]
\rm
\label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
+Fix a `target space' $T$, any topological space.
+We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of
all continuous maps from $X$ to $T$.
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo
@@ -506,14 +519,17 @@
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
-Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
+Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above.
+Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
\end{example}
\begin{example}[Maps to a space, with a fiber]
\rm
\label{ex:maps-to-a-space-with-a-fiber}%
We can modify the example above, by fixing a
-closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case.
+closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$,
+otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged.
+Taking $F$ to be a point recovers the previous case.
\end{example}
\begin{example}[Linearized, twisted, maps to a space]
@@ -530,7 +546,8 @@
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
\end{example}
-The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
+The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend.
+Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
\begin{example}[Traditional $n$-categories]
\rm
\label{ex:traditional-n-categories}
@@ -550,7 +567,10 @@
\nn{refer elsewhere for details?}
-Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.}
+Recall we described a system of fields and local relations based on a `traditional $n$-category'
+$C$ in Example \ref{ex:traditional-n-categories(fields)} above.
+Constructing a system of fields from $\cC$ recovers that example.
+\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.}
\end{example}
Finally, we describe a version of the bordism $n$-category suitable to our definitions.
@@ -593,7 +613,8 @@
\nn{maybe should also mention version where we enrich over spaces rather than chain complexes}
\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 $\pi^\infty_{\le n}(T)$.
+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 $\pi^\infty_{\le n}(T)$.
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
@@ -606,9 +627,19 @@
where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
\end{example}
-This example will be essential for Theorem \ref{product_thm} 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 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} 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.
+This example will be essential for Theorem \ref{product_thm} 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
+$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
+We think of this as providing a `free resolution' of the topological $n$-category.
+\todo{Say more here!}
+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.
-Be careful that the `free resolution' of the topological $n$-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. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
+Be careful that the `free resolution' of the topological $n$-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.
+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, $A_\infty$ version]
\rm
@@ -639,15 +670,30 @@
%\subsection{From $n$-categories to systems of fields}
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we describe how to extend an $n$-category $\cC$ as described above (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. This extension is a certain colimit, and we've chosen the notation to remind you of this.
+In this section we describe how to extend an $n$-category $\cC$ as described above
+(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
+This extension is a certain colimit, and we've chosen the notation to remind you of this.
That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension
-from $k$-balls to arbitrary $k$-manifolds. Recall that we've already anticipated this construction in the previous section, 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.
-In the case of plain $n$-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 $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', 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 $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$.
+from $k$-balls to arbitrary $k$-manifolds.
+Recall that we've already anticipated this construction in the previous section,
+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.
+In the case of plain $n$-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 $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
+Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution',
+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 $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$.
We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
-An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ 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 $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds 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).
+An $n$-category $\cC$ provides a functor from this poset to the category of sets,
+and we will define $\cC(W)$ 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 $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds 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).
\begin{defn}
Say that a `permissible decomposition' of $W$ is a cell decomposition
@@ -659,7 +705,8 @@
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$.
-The category $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
+The category $\cell(W)$ has objects the permissible decompositions of $W$,
+and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
See Figure \ref{partofJfig} for an example.
\end{defn}
@@ -695,7 +742,8 @@
When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a
closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and
-we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
+we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$.
+(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we
fix a field on $\bd W$
(i.e. fix an element of the colimit associated to $\bd W$).
@@ -710,12 +758,17 @@
\end{defn}
\begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ 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 $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
+When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ 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 $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
\end{defn}
-We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
+We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$
+with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
-We now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$
+We now give a more concrete description of the colimit in each case.
+If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold,
+we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$
\begin{equation*}
\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K
\end{equation*}
@@ -732,7 +785,9 @@
\[
V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
-where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.)
+where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
+(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$,
+the complex $U[m]$ is concentrated in degree $m$.)
We endow $V$ 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 $\bar{x}$
@@ -752,7 +807,8 @@
permissible decomposition (filtration degree 0).
Then we glue these together with mapping cylinders coming from gluing maps
(filtration degree 1).
-Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on.
+Then we kill the extra homology we just introduced with mapping
+cylinders between the mapping cylinders (filtration degree 2), and so on.
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
@@ -781,7 +837,9 @@
\nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.}
-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.
+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.
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
@@ -819,7 +877,8 @@
the category of marked $k$-hemispheres and
homeomorphisms to the category of sets and bijections.}
\end{lem}
-The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. We use the same type of colimit construction.
+The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details.
+We use the same type of colimit construction.
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
@@ -1040,13 +1099,22 @@
\end{example}
\begin{example}
-Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
+Suppose $S$ is a topological space, with a subspace $T$.
+We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$
+for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs
+$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all
+such maps modulo homotopies fixed on $\bdy B \setminus N$.
+This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and
+\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to
+Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
\end{example}
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
-Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$),
+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 $Y_i$.
@@ -1055,7 +1123,8 @@
%component $\bd_i W$ of $W$.
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.)
-We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above.
+We will define a set $\cC(W, \cN)$ using a colimit construction 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.)
@@ -1070,7 +1139,8 @@
\begin{figure}[!ht]\begin{equation*}
\mathfig{.4}{ncat/mblabel}
\end{equation*}\caption{A permissible decomposition of a manifold
-whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
+whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.
+Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
This defines a partial ordering $\cell(W)$, which we will think of as a category.
@@ -1096,7 +1166,8 @@
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 $D \mapsto \cC(D\times W, \cN)$
+$D\times Y_i \sub \bd(D\times W)$.
+It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$.
\medskip
@@ -1110,7 +1181,8 @@
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
-$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially)
+$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$.
+This of course depends (functorially)
on the choice of 1-ball $J$.
We will define a more general self tensor product (categorified coend) below.
@@ -1132,7 +1204,8 @@
we need to define morphisms of $A_\infty$ $1$-category modules and establish
some of their elementary properties.
-To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
+To motivate the definitions which follow, consider algebras $A$ and $B$,
+right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
\begin{eqnarray*}
\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\
f &\mapsto& [x \mapsto f(x\ot -)] \\
@@ -1260,7 +1333,9 @@
\[
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) .
\]
-For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals
+For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$,
+where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and
+$\cbar''$ corresponds to the subintervals
which are dropped off the right side.
(Either $\cbar'$ or $\cbar''$ might be empty.)
\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.}
@@ -1389,8 +1464,10 @@
\label{feb21a}
\end{figure}
-The $0$-marked balls can be cut into smaller balls in various ways. We only consider those decompositions in which the smaller balls are either
- $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) or plain (don't intersect the $0$-marking of the large ball).
+The $0$-marked balls can be cut into smaller balls in various ways.
+We only consider those decompositions in which the smaller balls are either
+$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc)
+or plain (don't intersect the $0$-marking of the large ball).
We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere.
Fix $n$-categories $\cA$ and $\cB$.