blob: comparison text/ncat.tex

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-:1c408505c9f5
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 \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.}
 %\nn{should they be called $n$-modules instead of just modules?  probably not, but worth considering.}
 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
 in the context of an $m{+}1$-dimensional TQFT.
 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
 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
 \subsection{Morphisms of $A_\infty$ 1-cat modules}
 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
 \begin{eqnarray*}
 	\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\
 and modules $\cM_\cC$ and $_\cC\cN$,
 \[
 	(\cM_\cC\ot {_\cC\cN})^* \cong  \hom_\cC(\cM_\cC \to (_\cC\cN)^*) .
 \]
-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
 $\hom_\cC$.
 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules
 for general $n$.
 	(_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* ,
 \]
 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 $(\cM_\cC\ot {_\cC\cN})^*$
 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$.
 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$.
 Subdivisions are just the obvious thing, but antirefinements are defined to mimic
 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always
 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
 \[
 	\prod_l \prod_{\olD} \hom[l]\left(
 	g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1})
 \]
 constitutes a null homotopy of
 $g((\bd \olD)\ot -)$ (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?}

changeset 262	3278eafef668
parent 261	1c408505c9f5
child 265	73c62576ef70