720 |
721 |
721 \subsection{Modules} |
722 \subsection{Modules} |
722 |
723 |
723 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
724 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
724 a.k.a.\ actions). |
725 a.k.a.\ actions). |
725 The definition will be very similar to that of $n$-categories. |
726 The definition will be very similar to that of $n$-categories, |
|
727 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
726 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
728 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
727 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
729 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
728 |
|
729 \nn{** resume revising here} |
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730 |
730 |
731 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
731 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
732 in the context of an $m{+}1$-dimensional TQFT. |
732 in the context of an $m{+}1$-dimensional TQFT. |
733 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
733 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
734 This will be explained in more detail as we present the axioms. |
734 This will be explained in more detail as we present the axioms. |
739 (standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
739 (standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
740 We call $B$ the ball and $N$ the marking. |
740 We call $B$ the ball and $N$ the marking. |
741 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
741 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
742 restricts to a homeomorphism of markings. |
742 restricts to a homeomorphism of markings. |
743 |
743 |
744 \xxpar{Module morphisms} |
744 \mmpar{Module axiom 1}{Module morphisms} |
745 {For each $0 \le k \le n$, we have a functor $\cM_k$ from |
745 {For each $0 \le k \le n$, we have a functor $\cM_k$ from |
746 the category of marked $k$-balls and |
746 the category of marked $k$-balls and |
747 homeomorphisms to the category of sets and bijections.} |
747 homeomorphisms to the category of sets and bijections.} |
748 |
748 |
749 (As with $n$-categories, we will usually omit the subscript $k$.) |
749 (As with $n$-categories, we will usually omit the subscript $k$.) |
762 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
762 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
763 |
763 |
764 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
764 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
765 Call such a thing a {marked $k{-}1$-hemisphere}. |
765 Call such a thing a {marked $k{-}1$-hemisphere}. |
766 |
766 |
767 \xxpar{Module boundaries, part 1:} |
767 \mmpar{Module axiom 2}{Module boundaries (hemispheres)} |
768 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
768 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
769 the category of marked $k$-hemispheres and |
769 the category of marked $k$-hemispheres and |
770 homeomorphisms to the category of sets and bijections.} |
770 homeomorphisms to the category of sets and bijections.} |
771 |
771 |
772 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
772 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
773 |
773 |
774 \xxpar{Module boundaries, part 2:} |
774 \mmpar{Module axiom 3}{Module boundaries (maps)} |
775 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
775 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
776 These maps, for various $M$, comprise a natural transformation of functors.} |
776 These maps, for various $M$, comprise a natural transformation of functors.} |
777 |
777 |
778 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
778 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
779 |
779 |
780 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
780 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
781 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
781 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
782 and $c\in \cC(\bd M)$. |
782 and $c\in \cC(\bd M)$. |
783 |
783 |
784 \xxpar{Module domain $+$ range $\to$ boundary:} |
784 \mmpar{Module axiom 4}{Boundary from domain and range} |
785 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
785 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
786 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
786 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
787 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
787 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
788 two maps $\bd: \cM(M_i)\to \cM(E)$. |
788 two maps $\bd: \cM(M_i)\to \cM(E)$. |
789 Then (axiom) we have an injective map |
789 Then (axiom) we have an injective map |
790 \[ |
790 \[ |
791 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
791 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
792 \] |
792 \] |
793 which is natural with respect to the actions of homeomorphisms.} |
793 which is natural with respect to the actions of homeomorphisms.} |
794 |
794 |
795 Let $\cM(H)_E$ denote the image of $\gl_E$. |
795 Let $\cM(H)_E$ denote the image of $\gl_E$. |
796 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
796 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
797 |
797 |
798 |
798 |
799 \xxpar{Axiom yet to be named:} |
799 \mmpar{Module axiom 5}{Module to category restrictions} |
800 {For each marked $k$-hemisphere $H$ there is a restriction map |
800 {For each marked $k$-hemisphere $H$ there is a restriction map |
801 $\cM(H)\to \cC(H)$. |
801 $\cM(H)\to \cC(H)$. |
802 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
802 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
803 These maps comprise a natural transformation of functors.} |
803 These maps comprise a natural transformation of functors.} |
804 |
804 |