# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1278522468 21600
# Node ID d55b85632926ba8439ffbe4c4a7949c0505c7f6d
# Parent  a896ec29425496f783371d8c7b48af9774a0b128
more ncat (colimits)

diff -r a896ec294254 -r d55b85632926 text/ncat.tex
--- a/text/ncat.tex	Wed Jul 07 10:17:30 2010 -0600
+++ b/text/ncat.tex	Wed Jul 07 11:07:48 2010 -0600
@@ -993,11 +993,19 @@
 We can specify boundary data $c \in \cl{\cC}(\bdy W)$, 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 $\cl{\cC}(W,c)$ to be the direct sum over all permissible decompositions of $W$
+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
+\[
+	\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim ,
+\]
+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 \psi_{\cC;W,c}(x)\right) \big/ K
+	\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / 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)
@@ -1015,6 +1023,7 @@
 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 $m$.)
+\nn{if there is a std convention, should we use it?  or are we deliberately bucking tradition?}
 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 $\bar{x}$
@@ -1024,12 +1033,13 @@
 \]
 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$
 is the usual gluing map coming from the antirefinement $x_0 \le x_1$.
-\nn{need to say this better}
-\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which
-combine only two balls at a time; for $n=1$ this version will lead to usual definition
-of $A_\infty$ category}
+%\nn{need to say this better}
+%\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which
+%combine only two balls at a time; for $n=1$ this version will lead to usual definition
+%of $A_\infty$ category}
 
 We will call $m$ the filtration degree of the complex.
+\nn{is there a more standard term for this?}
 We can think of this construction as starting with a disjoint copy of a complex for each
 permissible decomposition (filtration degree 0).
 Then we glue these together with mapping cylinders coming from gluing maps
@@ -1037,10 +1047,10 @@
 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. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
+$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
 
 It is easy to see that
-there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
+there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps
 comprise a natural transformation of functors.
 
 \begin{lem}