567 

   567 \nn{start with (less general) tensor products; maybe change this later}

   568 

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

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

   571 

   572 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$.

   573 (If $k=1$ and manifolds are oriented, then one should be 

   574 a left module and the other a right module.)

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

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

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

   578 $\cM\ot_\cC\cM'(D)$ will have additional structure; see below.)

   579 $\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$,

   580 which we define next.

   581 

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

   583 \[

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

   585 \]

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

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

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

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

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

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

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

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

   594 \nn{need figures}

   595 

   596 $\cC$, $\cM$ and $\cM'$ determine 

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

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

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

   600 \[

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

   602 \]

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

   604 $X_a$, $M_b$ and $M'_c$ all agree.

   605 (Think fibered product.)

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

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

   608 

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

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

   611 $\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps

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

   613 

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

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

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

   617 \nn{say this more precisely?}