
diff -r ec3af8dfcb3c -r a3311a926113 text/ncat.tex
--- a/text/ncat.tex	Tue Jul 21 16:21:20 2009 +0000
+++ b/text/ncat.tex	Tue Jul 21 18:45:18 2009 +0000
@@ -253,7 +253,23 @@
 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex
 type $A_\infty$ $n$-category.
 
+\medskip
 
+The alert reader will have already noticed that our definition of (plain) $n$-category
+is extremely similar to our definition of topological fields.
+The only difference is that for the $n$-category definition we restrict our attention to balls
+(and their boundaries), while for fields we consider all manifolds.
+\nn{also: difference at the top dimension; fix this}
+Thus a system of fields determines an $n$-category simply by restricting our attention to
+balls.
+The $n$-category can be thought of as the local part of the fields.
+Conversely, given an $n$-category we can construct a system of fields via 
+\nn{gluing, or a universal construction}
+
+\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems
+of fields.
+The universal (colimit) construction becomes our generalized definition of blob homology.
+Need to explain how it relates to the old definition.}
 
 \medskip
 
@@ -275,9 +291,10 @@
 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat
 \item conversely, our def implies other defs
 \item traditional $A_\infty$ 1-cat def implies our def
-\item ... and vice-versa
+\item ... and vice-versa (already done in appendix)
 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?)
 \item spell out what difference (if any) Top vs PL vs Smooth makes
+\item explain relation between old-fashioned blob homology and new-fangled blob homology
 \end{itemize}
 
 




