--- a/text/ncat.tex Tue Jan 05 21:18:39 2010 +0000
+++ b/text/ncat.tex Sun Jan 10 18:27:49 2010 +0000
@@ -593,9 +593,11 @@
\subsection{From $n$-categories to systems of fields}
\label{ss:ncat_fields}
-In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variation) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds.
+In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds.
-We will first define the `cell-decomposition' poset $\cJ(W)$ 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) 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 complex to $n$-manifolds with boundary data), then the resulting system of fields 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).
+We will first define the `cell-decomposition' poset $\cJ(W)$ 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 system of fields 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
@@ -621,27 +623,32 @@
-\nn{resume revising here}
-
An $n$-category $\cC$ determines
a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets
(possibly with additional structure if $k=n$).
-For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
+Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
+and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
+are splittable along this decomposition.
+%For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
\begin{defn}
Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows.
For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset
\begin{equation}
\label{eq:psi-C}
- \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)_{\bdy X_a}
+ \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
$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
\end{defn}
-When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is an $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$
+When the $n$-category $\cC$ is enriched in some 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.)
+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 $\bd W$).
Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
@@ -658,7 +665,7 @@
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 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*}
@@ -666,8 +673,10 @@
$a\in \psi_{\cC;W,c}(x)$ 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 $x \leq y$.
+\nn{ ** resume revising here}
+
In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit
-is slightly more involved.
+is more involved.
%\nn{should probably rewrite this to be compatible with some standard reference}
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.
Such sequences (for all $m$) form a simplicial set in $\cJ(W)$.