# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1308235900 21600
# Node ID 4d66ffe8dc85885ad5526eec8f993dc1c7b8641b
# Parent  24f14faacab4282081d5088011400d914fd6446d
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure

diff -r 24f14faacab4 -r 4d66ffe8dc85 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Wed Jun 15 14:15:19 2011 -0600
+++ b/text/appendixes/famodiff.tex	Thu Jun 16 08:51:40 2011 -0600
@@ -258,7 +258,8 @@
 \item $h(p, 0) = f(p)$ for all $p\in P$.
 \item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$,
 for all $p\in P$.
-\item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on $U_i^{i-1}$
+\item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on 
+$U_i^{i-1} \setmin (U_0^i \cup \cdots \cup U_{i-1}^i)$
 (and hence supported on $U_i$).
 \end{itemize}
 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small,
diff -r 24f14faacab4 -r 4d66ffe8dc85 text/ncat.tex
--- a/text/ncat.tex	Wed Jun 15 14:15:19 2011 -0600
+++ b/text/ncat.tex	Thu Jun 16 08:51:40 2011 -0600
@@ -676,8 +676,8 @@
 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then 
 $\cC(Y; c)$ is just a plain set.
 
-We will aim for a little bit more generality than we need and not assume that the objects
-of our auxiliary category are sets with extra structure.
+%We will aim for a little bit more generality than we need and not assume that the objects
+%of our auxiliary category are sets with extra structure.
 First we must specify requirements for the auxiliary category.
 It should have a {\it distributive monoidal structure} in the sense of 
 \nn{Stolz and Teichner, Traces in monoidal categories, 1010.4527}.
@@ -688,6 +688,9 @@
 \item vector spaces (or $R$-modules or chain complexes) with tensor product and direct sum; and
 \item topological spaces with product and disjoint union.
 \end{itemize}
+For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure.
+(Otherwise, stating the axioms for identity morphisms becomes more cumbersome.)
+
 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
 we need a preliminary definition.
 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the 
@@ -712,7 +715,7 @@
 \]
 where the sum is over $c\in\cC(Y)$ such that $\bd c = d$.
 This map is natural with respect to the action of homeomorphisms and with respect to restrictions.
-\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.}
+%\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.}
 \end{itemize}
 \end{axiom}