5 |
5 |
6 \section{$n$-categories and their modules} |
6 \section{$n$-categories and their modules} |
7 \label{sec:ncats} |
7 \label{sec:ncats} |
8 |
8 |
9 \subsection{Definition of $n$-categories} |
9 \subsection{Definition of $n$-categories} |
|
10 \label{ss:n-cat-def} |
10 |
11 |
11 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
13 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
13 (As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of |
14 (As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of |
14 a `weak' $n$-category with `strong duality'.) |
15 a `weak' $n$-category with `strong duality'.) |
534 \rm |
535 \rm |
535 \label{ex:traditional-n-categories} |
536 \label{ex:traditional-n-categories} |
536 Given a `traditional $n$-category with strong duality' $C$ |
537 Given a `traditional $n$-category with strong duality' $C$ |
537 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
538 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
538 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
539 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
539 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X)$ to finite linear |
540 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
540 combinations of $C$-labeled sub cell complexes of $X$ |
541 combinations of $C$-labeled sub cell complexes of $X$ |
541 modulo the kernel of the evaluation map. |
542 modulo the kernel of the evaluation map. |
542 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
543 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
543 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
544 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
544 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
545 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |