1978 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
1978 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
1979 \label{ssec:spherecat} |
1979 \label{ssec:spherecat} |
1980 |
1980 |
1981 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules". |
1981 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules". |
1982 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$, |
1982 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$, |
1983 and the $n{+}1$-morphisms are intertwinors. |
1983 and the $n{+}1$-morphisms are intertwiners. |
1984 With future applications in mind, we treat simultaneously the big category |
1984 With future applications in mind, we treat simultaneously the big category |
1985 of all $n$-categories and all sphere modules and also subcategories thereof. |
1985 of all $n$-categories and all sphere modules and also subcategories thereof. |
1986 When $n=1$ this is closely related to familiar $2$-categories consisting of |
1986 When $n=1$ this is closely related to familiar $2$-categories consisting of |
1987 algebras, bimodules and intertwiners (or a subcategory of that). |
1987 algebras, bimodules and intertwiners (or a subcategory of that). |
1988 The sphere module $n{+}1$-category is a natural generalization of the |
1988 The sphere module $n{+}1$-category is a natural generalization of the |
1989 algebra-bimodule-intertwinor 2-category to higher dimensions. |
1989 algebra-bimodule-intertwiner 2-category to higher dimensions. |
1990 |
1990 |
1991 Another possible name for this $n{+}1$-category is $n{+}1$-category of defects. |
1991 Another possible name for this $n{+}1$-category is $n{+}1$-category of defects. |
1992 The $n$-categories are thought of as representing field theories, and the |
1992 The $n$-categories are thought of as representing field theories, and the |
1993 $0$-sphere modules are codimension 1 defects between adjacent theories. |
1993 $0$-sphere modules are codimension 1 defects between adjacent theories. |
1994 In general, $m$-sphere modules are codimension $m{+}1$ defects; |
1994 In general, $m$-sphere modules are codimension $m{+}1$ defects; |
2592 |
2592 |
2593 \medskip |
2593 \medskip |
2594 |
2594 |
2595 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. |
2595 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. |
2596 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent |
2596 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent |
2597 objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors. |
2597 objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners. |
2598 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the |
2598 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the |
2599 $n{+}1$-category of sphere modules. |
2599 $n{+}1$-category of sphere modules. |
2600 |
2600 |
2601 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
2601 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
2602 dimensions 1 and $n+1$ (the middle dimensions come along for free). |
2602 dimensions 1 and $n+1$ (the middle dimensions come along for free). |
2622 \end{figure} |
2622 \end{figure} |
2623 |
2623 |
2624 |
2624 |
2625 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2625 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2626 between various compositions of these 2-morphisms and various identity 2-morphisms. |
2626 between various compositions of these 2-morphisms and various identity 2-morphisms. |
2627 Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated |
2627 Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated |
2628 to decorated circles. |
2628 to decorated circles. |
2629 Figure \ref{morita-fig-2} |
2629 Figure \ref{morita-fig-2} |
2630 \begin{figure}[t] |
2630 \begin{figure}[t] |
2631 $$\mathfig{.55}{tempkw/morita2}$$ |
2631 $$\mathfig{.55}{tempkw/morita2}$$ |
2632 \caption{Intertwinors for a Morita equivalence}\label{morita-fig-2} |
2632 \caption{intertwiners for a Morita equivalence}\label{morita-fig-2} |
2633 \end{figure} |
2633 \end{figure} |
2634 shows the intertwinors we need. |
2634 shows the intertwiners we need. |
2635 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
2635 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
2636 on the boundary. |
2636 on the boundary. |
2637 This is the 3-dimensional part of the data for the Morita equivalence. |
2637 This is the 3-dimensional part of the data for the Morita equivalence. |
2638 (Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} |
2638 (Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} |
2639 are the same (up to rotation), as are the $h$ and $g$ arrows.) |
2639 are the same (up to rotation), as are the $h$ and $g$ arrows.) |
2642 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
2642 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
2643 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
2643 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
2644 These are illustrated in Figure \ref{morita-fig-3}. |
2644 These are illustrated in Figure \ref{morita-fig-3}. |
2645 \begin{figure}[t] |
2645 \begin{figure}[t] |
2646 $$\mathfig{.65}{tempkw/morita3}$$ |
2646 $$\mathfig{.65}{tempkw/morita3}$$ |
2647 \caption{Identities for intertwinors}\label{morita-fig-3} |
2647 \caption{Identities for intertwiners}\label{morita-fig-3} |
2648 \end{figure} |
2648 \end{figure} |
2649 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. |
2649 Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. |
2650 |
2650 |
2651 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
2651 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
2652 part of the Morita equivalence. |
2652 part of the Morita equivalence. |
2653 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
2653 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
2654 labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. |
2654 labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. |
2655 The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must |
2655 The $n{+}1$-dimensional part of the equivalence is given by certain intertwiners, and these intertwiners must |
2656 be invertible and satisfy |
2656 be invertible and satisfy |
2657 identities corresponding to Morse cancellations in $n$-manifolds. |
2657 identities corresponding to Morse cancellations in $n$-manifolds. |
2658 |
2658 |
2659 \noop{ |
2659 \noop{ |
2660 One way of thinking of these conditions is as follows. |
2660 One way of thinking of these conditions is as follows. |