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 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
-\section{$n$-categories}
+\section{Definitions of $n$-categories}
 \label{sec:ncats}
 \subsection{Definition of $n$-categories}
 Before proceeding, we need more appropriate definitions of $n$-categories,
 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$),
 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary
 %component $\bd_i W$ of $W$.
 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.)
-We will define a set $\cC(W, \cN)$ using a colimit construction similar to above.
+We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above.
-\nn{give ref}
+(If $k = n$ and our $n$-categories are enriched, then
-(If $k = n$ and our $k$-categories are enriched, then
 $\cC(W, \cN)$ will have additional structure; see below.)
 Define a permissible decomposition of $W$ to be a decomposition
 \[
 	W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) ,
 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and
 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$,
 with $M_{ib}\cap Y_i$ being the marking.
 (See Figure \ref{mblabel}.)
 \begin{figure}[!ht]\begin{equation*}
-\mathfig{.6}{ncat/mblabel}
+\mathfig{.4}{ncat/mblabel}
 \end{equation*}\caption{A permissible decomposition of a manifold
 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
 This defines a partial ordering $\cJ(W)$, which we will think of as a category.
 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique
 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
-$\cN$ determines
+The collection of modules $\cN$ determines
 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets
 (possibly with additional structure if $k=n$).
 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset
 \[
 	\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right)
 \]
 such that the restrictions to the various pieces of shared boundaries amongst the
 $X_a$ and $M_{ib}$ all agree.
-(Think fibered product.)
+(That is, the fibered product over the boundary maps.)
 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
-Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$.
+We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$.
-(Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means
+(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means
 homotopy colimit.)
 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define
 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold
-$D\times Y_i \sub \bd(D\times W)$.
+$D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
+has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$.
-It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$
-has the structure of an $n{-}k$-category.
 \medskip
 We will use a simple special case of the above
 construction to define tensor products
 of modules.
 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$.
-(If $k=1$ and manifolds are oriented, then one should be
+(If $k=1$ and our manifolds are oriented, then one should be
 a left module and the other a right module.)
 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
-Define the tensor product of $\cM_1$ and $\cM_2$ to be the
+Define the tensor product $\cM_1 \tensor \cM_2$ to be the
-$n{-}1$-category $\cT(J, \cM_1, \cM_2)$,
+$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially)
-\[
-	\cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) .
-\]
-This of course depends (functorially)
 on the choice of 1-ball $J$.
 We will define a more general self tensor product (categorified coend) below.
 %\nn{what about self tensor products /coends ?}
 \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
+we need to define morphisms of $A_\infty$ 1-category modules and establish
 some of their elementary properties.
-To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules
+To motivate the definitions which follow, consider algebras $A$ and $B$,  right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
-$X_B$, $_BY_A$ 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)) \\
 	f &\mapsto& [x \mapsto f(x\ot -)] \\
 	{}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g .
 \end{eqnarray*}
 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 we define the things appearing in the above equation:
+In the next few paragraphs we define the objects 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
+\def\olD{{\overline D}}
+\def\cbar{{\bar c}}
+In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules
 for general $n$.
 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$
 and their gluings (antirefinements).
-(The tensor product will depend (functorially) on the choice of $J$.)
+(This tensor product depends functorially on the choice of $J$.)
-To a subdivision
+To a subdivision $D$
 \[
 	J = I_1\cup \cdots\cup I_p
 \]
 we associate the chain complex
 \[
-	\cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) .
+	\psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) .
 \]
-(If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.)
 To each antirefinement we associate a chain map using the composition law of $\cC$ and the
 module actions of $\cC$ on $\cM$ and $\cN$.
-\def\olD{{\overline D}}
-\def\cbar{{\bar c}}
 The underlying graded vector space of the homotopy colimit is
 \[
 	\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] ,
 \]
 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$
-runs through chains of antirefinements, and $[l]$ denotes a grading shift.
+runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift.
 We will denote an element of the summand indexed by $\olD$ by
 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$.
-The boundary map is given (ignoring signs) by
+The boundary map is given by
-\begin{eqnarray*}
+\begin{align*}
-	\bd(\olD\ot m\ot\cbar\ot n) &=& \olD\ot\bd(m\ot\cbar)\ot n + \olD\ot m\ot\cbar\ot \bd n + \\
+	\bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\
-			& & \;\;	(\bd_+ \olD)\ot m\ot\cbar\ot n + (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) ,
+	& \qquad + (-1)^l \olD\ot\bd(m\ot\cbar\ot n)
-\end{eqnarray*}
+\end{align*}
-where $\bd_+ \olD = \sum_{i>0} (D_0, \cdots \widehat{D_i} \cdots , D_l)$ (the part of the simplicial
+where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial
-boundary which retains $D_0$), $\bd_0 \olD = (D_1, \cdots , D_l)$,
+boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$,
 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$.
 $(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$:
 \[
 	\prod_l \prod_{\olD} (\psi(D_0)[l])^* ,
 \begin{eqnarray*}
 	(\bd f)(\olD\ot m\ot\cbar\ot n) &=& f(\olD\ot\bd(m\ot\cbar)\ot n) +
 													f(\olD\ot m\ot\cbar\ot \bd n) + \\
 			& & \;\;	f((\bd_+ \olD)\ot m\ot\cbar\ot n) + f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) .
 \end{eqnarray*}
-(Again, we are ignoring signs.)
+(Again, we are ignoring signs.) \nn{put signs in}
 Next we define the dual module $(_\cC\cN)^*$.
 This will depend on a choice of interval $J$, just as the tensor product did.
 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals
 to chain complexes.
 \end{eqnarray*}
 We are almost ready to give the definition of morphisms between arbitrary modules
 $\cX_\cC$ and $\cY_\cC$.
 Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$.
-To fix this, we define subdivisions are antirefinements of left-marked intervals.
+To fix this, we define subdivisions as antirefinements of left-marked intervals.
 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.

changeset 286	ff867bfc8e9c
parent 279	cb16992373be
child 288	6c1b3c954c7e