--- a/text/ncat.tex	Sat Aug 15 15:47:52 2009 +0000
+++ b/text/ncat.tex	Mon Aug 17 05:23:35 2009 +0000
@@ -444,14 +444,48 @@
 (Think fibered product.)
 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$
 via the composition maps of $\cC$.
+(If $\dim(W) = n$ then we need to also make use of the monoidal
+product in the enriching category.
+\nn{should probably be more explicit here})
 
 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$.
-In other words, for each decomposition $x$ there is a map
+In the plain (non-$A_\infty$) case, this means that
+for each decomposition $x$ there is a map
 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps
 above, and $\cC(W)$ is universal with respect to these properties.
-\nn{in A-inf case, need to say more}
+In the $A_\infty$ case, it means 
+\nn{.... need to check if there is a def in the literature before writing this down}
+
+More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take
+\[
+	\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K
+\]
+where $K$ is generated by all things of the form $a - g(a)$, where
+$a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x)
+\to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$.
 
-\nn{should give more concrete description (two cases)}
+In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit
+is as follows.
+Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_{m-1}$ of permissible decompositions.
+Such sequences (for all $m$) form a simplicial set.
+Let
+\[
+	V = \bigoplus_{(x_i)} \psi_\cC(x_0) ,
+\]
+where the sum is over all $m$-sequences and all $m$.
+We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$
+summands plus another term using the differential of the simplicial set of $m$-sequences.
+More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
+summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define
+\[
+	\bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) ,
+\]
+where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$
+is the usual map.
+\nn{need to say this better}
+\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which
+combine only two balls at a time; for $n=1$ this version will lead to usual definition
+of $A_\infty$ category}
 
 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
 
@@ -692,13 +726,12 @@
 
 
 \subsection{Modules as boundary labels}
+\label{moddecss}
 
 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 $W$.
 
-\nn{need to generalize to labeling codim 0 submanifolds of the boundary}
-
 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above.
 \nn{give ref}
 (If $k = n$ and our $k$-categories are enriched, then
@@ -736,18 +769,21 @@
 $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps
 above, and $\cC(W, \cN)$ is universal with respect to these properties.
 
+More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.
+\nn{need to say more?}
+
 \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.}
 
 \subsection{Tensor products}
 
-Next we consider tensor products (or, more generally, self tensor products
-or coends).
+Next we consider tensor products.
+
+\nn{what about self tensor products /coends ?}
 
 \nn{maybe ``tensor product" is not the best name?}
 
 \nn{start with (less general) tensor products; maybe change this later}
 
-** \nn{stuff below needs to be rewritten (shortened), because of new subsections above}
 
 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$.
 (If $k=1$ and manifolds are oriented, then one should be 
@@ -755,97 +791,17 @@
 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially)
 on a choice of 1-ball (interval) $J$.
 
-
-
-
-
-
-Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball
-and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$.
-
-Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$.
-We will define a set $\cM\ot_\cC\cM'(D)$.
-(If $k = n$ and our $k$-categories are enriched, then
-$\cM\ot_\cC\cM'(D)$ will have additional structure; see below.)
-$\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$,
-which we define next.
-
-Define a permissible decomposition of $D$ to be a decomposition
+Let $p$ and $p'$ be the boundary points of $J$.
+Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with
+$X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}.
+Let
 \[
-	D = (\cup_a X_a) \cup (\cup_b M_b) \cup (\cup_c M'_c) ,
+	\cT(X) \deq \cC(X\times J, \cM, \cM') ,
 \]
-Where each $X_a$ is a plain $k$-ball (disjoint from the markings $N$ and $N'$ of $D$),
-each $M_b$ is a marked $k$-ball intersecting $N$, and
-each $M'_b$ is a marked $k$-ball intersecting $N'$.
-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 $\cJ(D)$, which we will think of as a category.
-(The objects of $\cJ(D)$ are permissible decompositions of $D$, and there is a unique
-morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
-\nn{need figures}
-
-$\cC$, $\cM$ and $\cM'$ determine 
-a functor $\psi$ from $\cJ(D)$ to the category of sets 
-(possibly with additional structure if $k=n$).
-For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset
-\[
-	\psi(x) \sub (\prod_a \cC(X_a)) \times (\prod_b \cM(M_b)) \times (\prod_c \cM'(M'_c))
-\]
-such that the restrictions to the various pieces of shared boundaries amongst the
-$X_a$, $M_b$ and $M'_c$ all agree.
-(Think fibered product.)
-If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$
-via the gluing (composition or action) maps from $\cC$, $\cM$ and $\cM'$.
+where the right hand side is the colimit construction defined in Subsection \ref{moddecss}.
+It is not hard to see that $\cT$ becomes an $n{-}1$-category.
+\nn{maybe follows from stuff (not yet written) in previous subsection?}
 
-Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$.
-In other words, for each decomposition $x$ there is a map
-$\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps
-above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties.
-
-Define a {\it marked $k$-annulus} to be a manifold homeomorphic
-to $S^{k-1}\times I$, with its entire boundary ``marked".
-Define the boundary of a doubly marked $k$-ball $(B, N, N')$ to be the marked
-$k{-}1$-annulus $\bd B \setmin(N\cup N')$.
-
-Using a colimit construction similar to the one above, we can define a set
-$\cM\ot_\cC\cM'(A)$ for any marked $k$-annulus $A$ (for $k < n$).
-
-$\cM\ot_\cC\cM'$ is (among other things) a functor from the category of 
-doubly marked $k$-balls ($k\le n$) and homeomorphisms to the category of sets.
-We have other functors, also denoted $\cM\ot_\cC\cM'$, from the category of 
-marked $k$-annuli ($k < n$) and homeomorphisms to the category of sets.
-
-For each marked $k$-ball $D$ there is a restriction map
-\[
-	\bd : \cM\ot_\cC\cM(D) \to \cM\ot_\cC\cM(\bd D) .
-\]
-These maps comprise a natural transformation of functors.
-\nn{possible small problem: might need to define $\cM$ of a singly marked annulus}
-
-For $c \in \cM\ot_\cC\cM(\bd D)$, let 
-\[
-	\cM\ot_\cC\cM(D; c) \deq \bd\inv(c) .
-\]
-
-Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$,
-then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category
-(e.g.\ vector space or chain complex).
-
-Let $J$ be a doubly marked 1-ball (i.e. an interval, where we think of both endpoints
-as marked).
-For $X$ a plain $k$-ball ($k \le n-1$) or $k$-sphere ($k \le n-2$), define
-\[
-	\cM\ot_\cC\cM'(X) \deq \cM\ot_\cC\cM'(X\times J) .
-\]
-We claim that $\cM\ot_\cC\cM'$ has the structure of an $n{-}1$-category.
-We have already defined restriction maps $\bd : \cM\ot_\cC\cM'(X) \to 
-\cM\ot_\cC\cM'(\bd X)$.
-The only data for the $n{-}1$-category that we have not defined yet are the product
-morphisms.
-\nn{so next define those}
-
-\nn{need to check whether any of the steps in verifying that we have
-an $n{-}1$-category are non-trivial.}