449 In other words, for each decomposition $x$ there is a map

   452 In the plain (non-$A_\infty$) case, this means that

   453 for each decomposition $x$ there is a map

   452 \nn{in A-inf case, need to say more}

   456 In the $A_\infty$ case, it means 

   453 

   457 \nn{.... need to check if there is a def in the literature before writing this down}

   454 \nn{should give more concrete description (two cases)}

   458 

   459 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take

   460 \[

   461 	\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K

   462 \]

   463 where $K$ is generated by all things of the form $a - g(a)$, where

   464 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x)

   465 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$.

   466 

   467 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit

   468 is as follows.

   469 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_{m-1}$ of permissible decompositions.

   470 Such sequences (for all $m$) form a simplicial set.

   471 Let

   472 \[

   473 	V = \bigoplus_{(x_i)} \psi_\cC(x_0) ,

   474 \]

   475 where the sum is over all $m$-sequences and all $m$.

   476 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$

   477 summands plus another term using the differential of the simplicial set of $m$-sequences.

   478 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$

   479 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define

   480 \[

   481 	\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})) ,

   482 \]

   483 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$

   484 is the usual map.

   485 \nn{need to say this better}

   486 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which

   487 combine only two balls at a time; for $n=1$ this version will lead to usual definition

   488 of $A_\infty$ category}

   743 Next we consider tensor products (or, more generally, self tensor products

   779 Next we consider tensor products.

   744 or coends).

   780 

   781 \nn{what about self tensor products /coends ?}

   758 

   794 Let $p$ and $p'$ be the boundary points of $J$.

   759 

   795 Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with

   760 

   796 $X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}.

   761 

   797 Let

   762 

   798 \[

   763 Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball

   799 	\cT(X) \deq \cC(X\times J, \cM, \cM') ,

   764 and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$.

   800 \]

   765 

   801 where the right hand side is the colimit construction defined in Subsection \ref{moddecss}.

   766 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$.

   802 It is not hard to see that $\cT$ becomes an $n{-}1$-category.

   767 We will define a set $\cM\ot_\cC\cM'(D)$.

   803 \nn{maybe follows from stuff (not yet written) in previous subsection?}

   768 (If $k = n$ and our $k$-categories are enriched, then

   804 

   769 $\cM\ot_\cC\cM'(D)$ will have additional structure; see below.)

   770 $\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$,

   771 which we define next.

   772 

   773 Define a permissible decomposition of $D$ to be a decomposition

   774 \[

   775 	D = (\cup_a X_a) \cup (\cup_b M_b) \cup (\cup_c M'_c) ,

   776 \]

   777 Where each $X_a$ is a plain $k$-ball (disjoint from the markings $N$ and $N'$ of $D$),

   778 each $M_b$ is a marked $k$-ball intersecting $N$, and

   779 each $M'_b$ is a marked $k$-ball intersecting $N'$.

   780 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement

   781 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.

   782 This defines a partial ordering $\cJ(D)$, which we will think of as a category.

   783 (The objects of $\cJ(D)$ are permissible decompositions of $D$, and there is a unique

   784 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)

   785 \nn{need figures}

   786 

   787 $\cC$, $\cM$ and $\cM'$ determine 

   788 a functor $\psi$ from $\cJ(D)$ to the category of sets 

   789 (possibly with additional structure if $k=n$).

   790 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset

   791 \[

   792 	\psi(x) \sub (\prod_a \cC(X_a)) \times (\prod_b \cM(M_b)) \times (\prod_c \cM'(M'_c))

   793 \]

   794 such that the restrictions to the various pieces of shared boundaries amongst the

   795 $X_a$, $M_b$ and $M'_c$ all agree.

   796 (Think fibered product.)

   797 If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$

   798 via the gluing (composition or action) maps from $\cC$, $\cM$ and $\cM'$.

   799 

   800 Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$.

   801 In other words, for each decomposition $x$ there is a map

   802 $\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps

   803 above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties.

   804 

   805 Define a {\it marked $k$-annulus} to be a manifold homeomorphic

   806 to $S^{k-1}\times I$, with its entire boundary ``marked".

   807 Define the boundary of a doubly marked $k$-ball $(B, N, N')$ to be the marked

   808 $k{-}1$-annulus $\bd B \setmin(N\cup N')$.

   809 

   810 Using a colimit construction similar to the one above, we can define a set

   811 $\cM\ot_\cC\cM'(A)$ for any marked $k$-annulus $A$ (for $k < n$).

   812 

   813 $\cM\ot_\cC\cM'$ is (among other things) a functor from the category of 

   814 doubly marked $k$-balls ($k\le n$) and homeomorphisms to the category of sets.

   815 We have other functors, also denoted $\cM\ot_\cC\cM'$, from the category of 

   816 marked $k$-annuli ($k < n$) and homeomorphisms to the category of sets.

   817 

   818 For each marked $k$-ball $D$ there is a restriction map

   819 \[

   820 	\bd : \cM\ot_\cC\cM(D) \to \cM\ot_\cC\cM(\bd D) .

   821 \]

   822 These maps comprise a natural transformation of functors.

   823 \nn{possible small problem: might need to define $\cM$ of a singly marked annulus}

   824 

   825 For $c \in \cM\ot_\cC\cM(\bd D)$, let 

   826 \[

   827 	\cM\ot_\cC\cM(D; c) \deq \bd\inv(c) .

   828 \]

   829 

   830 Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$,

   831 then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category

   832 (e.g.\ vector space or chain complex).

   833 

   834 Let $J$ be a doubly marked 1-ball (i.e. an interval, where we think of both endpoints

   835 as marked).

   836 For $X$ a plain $k$-ball ($k \le n-1$) or $k$-sphere ($k \le n-2$), define

   837 \[

   838 	\cM\ot_\cC\cM'(X) \deq \cM\ot_\cC\cM'(X\times J) .

   839 \]

   840 We claim that $\cM\ot_\cC\cM'$ has the structure of an $n{-}1$-category.

   841 We have already defined restriction maps $\bd : \cM\ot_\cC\cM'(X) \to 

   842 \cM\ot_\cC\cM'(\bd X)$.

   843 The only data for the $n{-}1$-category that we have not defined yet are the product

   844 morphisms.

   845 \nn{so next define those}

   846 

   847 \nn{need to check whether any of the steps in verifying that we have

   848 an $n{-}1$-category are non-trivial.}