
diff -r 1c408505c9f5 -r 3278eafef668 text/ncat.tex
--- a/text/ncat.tex	Mon May 10 19:34:59 2010 -0700
+++ b/text/ncat.tex	Wed May 12 15:57:20 2010 -0700
@@ -744,7 +744,7 @@
 
 \subsection{Modules}
 
-Next we define topological and $A_\infty$ $n$-category modules.
+Next we define plain and $A_\infty$ $n$-category modules.
 The definition will be very similar to that of $n$-categories,
 but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.}
@@ -755,6 +755,8 @@
 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
 This will be explained in more detail as we present the axioms.
 
+\nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.}
+
 Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases.
 
 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
@@ -1096,7 +1098,7 @@
 In order to state and prove our version of the higher dimensional Deligne conjecture
 (Section \ref{sec:deligne}),
 we need to define morphisms of $A_\infty$ 1-cat modules and establish
-some elementary properties of these.
+some of their elementary properties.
 
 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules
 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction
@@ -1116,7 +1118,7 @@
 	(\cM_\cC\ot {_\cC\cN})^* \cong  \hom_\cC(\cM_\cC \to (_\cC\cN)^*) .
 \]
 
-In the next few paragraphs define the things appearing in the above equation:
+In the next few paragraphs we define the things appearing in the above equation:
 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally
 $\hom_\cC$.
 
@@ -1179,6 +1181,7 @@
 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated
 to the right-marked interval $J\setmin K$.
 This extends to a functor from all left-marked intervals (not just those contained in $J$).
+\nn{need to say more here; not obvious how homeomorphisms act}
 It's easy to verify the remaining module axioms.
 
 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$
@@ -1201,7 +1204,10 @@
 omitted.
 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by 
 gluing subintervals together and/or omitting some of the rightmost subintervals.
-(See Figure xxxx.)
+(See Figure \ref{fig:lmar}.)
+\begin{figure}[t]\begin{equation*}
+\mathfig{.6}{tempkw/left-marked-antirefinements}
+\end{equation*}\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure}
 
 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$.
 The underlying vector space is 
@@ -1243,6 +1249,47 @@
 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$
 should be interpreted as above).
 
+Define a {\it naive morphism} 
+\nn{should consider other names for this}
+of modules to be a collection of {\it chain} maps
+\[
+	h_K : \cX(K)\to \cY(K)
+\]
+for each left-marked interval $K$.
+These are required to commute with gluing;
+for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes:
+\[ \xymatrix{
+	\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} 
+							\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) 
+								\ar[d]^{\gl} \\
+	\cX(K) \ar[r]^{h_{K}} & \cY(K)
+} \]
+Given such an $h$ we can construct a non-naive morphism $g$, with $\bd g = 0$, as follows.
+Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0.
+If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define
+\[
+	g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) .
+\]
+Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$.
+If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing.
+If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact 
+that each $h_K$ is a chain map.
+
+\medskip
+
+Given $_\cC\cZ$ and  $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map
+\[
+	g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ .
+\]
+\nn{this is fairly straightforward, but the details are messy enough that I'm inclined
+to postpone writing it up, in the hopes that I'll think of a better way to organize things.}
+
+
+
+
+\medskip
+
+
 \nn{do we need to say anything about composing morphisms of modules?}
 
 \nn{should we define functors between $n$-cats in a similar way?}




