27 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball.

    29 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to 

    30 the standard $k$-ball.

   354 \medskip

   357 

   358 

   359 

   360 

   361 

   362 \subsection{From $n$-categories to systems of fields}

   363 

   364 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows.

   365 

   366 Let $W$ be a $k$-manifold, $1\le k \le n$.

   367 We will define a set $\cC(W)$.

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

   369 $\cC(W)$ will have additional structure; see below.)

   370 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$,

   371 which we define next.

   372 

   373 Define a permissible decomposition of $W$ to be a decomposition

   374 \[

   375 	W = \bigcup_a X_a ,

   376 \]

   377 where each $X_a$ is a $k$-ball.

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

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

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

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

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

   383 \nn{need figures}

   384 

   385 $\cC$ determines 

   386 a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets 

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

   388 For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset

   389 \[

   390 	\psi_\cC(x) \sub \prod_a \cC(X_a)

   391 \]

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

   393 $X_a$ all agree.

   394 (Think fibered product.)

   395 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$

   396 via the composition maps of $\cC$.

   397 

   398 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$.

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

   400 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps

   401 above, and $\cC(W)$ is universal with respect to these properties.

   402 

   403 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.

   404 

   405 It is easy to see that

   406 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps

   407 comprise a natural transformation of functors.

   408 

   409 \nn{need to finish explaining why we have a system of fields;

   410 need to say more about ``homological" fields? 

   411 (actions of homeomorphisms);

   412 define $k$-cat $\cC(\cdot\times W)$}

   413 

   414 

   415 

   416 \subsection{Modules}

   562 \medskip

   624 

   625 \subsection{Modules as boundary labels}

   626 

   627 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$),

   628 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary 

   629 component $\bd_i W$ of $W$.

   630 

   631 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above.

   632 \nn{give ref}

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

   634 $\cC(W, \cN)$ will have additional structure; see below.)

   635 

   636 Define a permissible decomposition of $W$ to be a decomposition

   637 \[

   638 	W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) ,

   639 \]

   640 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and

   641 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$,

   642 with $M_{ib}\cap\bd_i W$ being the marking.

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

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

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

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

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

   648 \nn{need figures}

   649 

   650 $\cN$ determines 

   651 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets 

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

   653 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset

   654 \[

   655 	\psi_\cN(x) \sub (\prod_a \cC(X_a)) \prod (\prod_{ib} \cN_i(M_{ib}))

   656 \]

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

   658 $X_a$ and $M_{ib}$ all agree.

   659 (Think fibered product.)

   660 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$

   661 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.

   662 

   663 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$.

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

   665 $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps

   666 above, and $\cC(W, \cN)$ is universal with respect to these properties.

   667 

   668 \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.}

   669 

   670 \subsection{Tensor products}