
diff -r 3df8116f1f0a -r 57425531f564 text/ncat.tex
--- a/text/ncat.tex	Sat Jun 25 06:27:16 2011 -0700
+++ b/text/ncat.tex	Sat Jun 25 06:44:35 2011 -0700
@@ -34,10 +34,11 @@
 
 The axioms for an $n$-category are spread throughout this section.
 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, 
-\ref{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product} and 
-\ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace 
+\ref{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and 
+\ref{axiom:extended-isotopies}.
+For an enriched $n$-category we add \ref{axiom:enriched}.
+For an $A_\infty$ $n$-category, we replace 
 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
-\nn{need to revise this after we're done rearranging the a-inf and enriched stuff}
 
 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms 
 for $k{-}1$-morphisms.
@@ -984,14 +985,16 @@
 In the $n$-category axioms above we have intermingled data and properties for expository reasons.
 Here's a summary of the definition which segregates the data from the properties.
 
-An $n$-category consists of the following data: \nn{need to revise this list}
+An $n$-category consists of the following data:
 \begin{itemize}
 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition});
 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product});
-\item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$;
-\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}).
+\item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched});
+%\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}).
+\item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions
+and collar maps (Axiom \ref{axiom:families}).
 \end{itemize}
 The above data must satisfy the following conditions:
 \begin{itemize}
@@ -1001,8 +1004,10 @@
 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}).
 \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}).
 \item If enriching in an auxiliary category, all of the data should be compatible 
-with the auxiliary category structure on $\cC_n(X; c)$.
-\item For ordinary categories, invariance of $n$-morphisms under extended isotopies (Axiom \ref{axiom:extended-isotopies}).
+with the auxiliary category structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}).
+\item The possible splittings of a morphism satisfy various conditions (Axiom \ref{axiom:vcones}).
+\item For ordinary categories, invariance of $n$-morphisms under extended isotopies 
+and collar maps (Axiom \ref{axiom:extended-isotopies}).
 \end{itemize}
 
 




