2291 \label{moddecss} |
2291 \label{moddecss} |
2292 |
2292 |
2293 Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$. |
2293 Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$. |
2294 Let $W$ be a $k$-manifold ($k\le n$), |
2294 Let $W$ be a $k$-manifold ($k\le n$), |
2295 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
2295 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
2296 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
2296 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each $Y_i$. |
2297 |
2297 |
2298 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
2298 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
2299 the one appearing in \S \ref{ss:ncat_fields} above. |
2299 the one appearing in \S \ref{ss:ncat_fields} above. |
2300 (If $k = n$ and our $n$-categories are enriched, then |
2300 (If $k = n$ and our $n$-categories are enriched, then |
2301 $\cC(W, \cN)$ will have additional structure; see below.) |
2301 $\cC(W, \cN)$ will have additional structure; see below.) |
2329 \[ |
2329 \[ |
2330 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
2330 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
2331 \] |
2331 \] |
2332 such that the restrictions to the various pieces of shared boundaries amongst the |
2332 such that the restrictions to the various pieces of shared boundaries amongst the |
2333 $X_a$ and $M_{ib}$ all agree. |
2333 $X_a$ and $M_{ib}$ all agree. |
2334 (That is, the fibered product over the boundary restriction maps.) |
2334 %(That is, the fibered product over the boundary restriction maps.) |
2335 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
2335 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
2336 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
2336 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
2337 |
2337 |
2338 We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
2338 We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
2339 (As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever |
2339 (As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever |
2373 Modules are collections of functors together with some additional data, so we define morphisms |
2373 Modules are collections of functors together with some additional data, so we define morphisms |
2374 of modules to be collections of natural transformations which are compatible with this |
2374 of modules to be collections of natural transformations which are compatible with this |
2375 additional data. |
2375 additional data. |
2376 |
2376 |
2377 More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors |
2377 More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors |
2378 $\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets |
2378 $\{\cX_k\}$ and $\{\cY_k\}$, for $1\le k\le n$, from marked $k$-balls to sets |
2379 as in Module Axiom \ref{module-axiom-funct}. |
2379 as in Module Axiom \ref{module-axiom-funct}. |
2380 A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$ |
2380 A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$ |
2381 satisfying: |
2381 satisfying: |
2382 \begin{itemize} |
2382 \begin{itemize} |
2383 \item Each $g_k$ commutes with $\bd$. |
2383 \item Each $g_k$ commutes with $\bd$. |
2446 \label{ssec:spherecat} |
2446 \label{ssec:spherecat} |
2447 |
2447 |
2448 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules". |
2448 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules". |
2449 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$, |
2449 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$, |
2450 and the $n{+}1$-morphisms are intertwiners. |
2450 and the $n{+}1$-morphisms are intertwiners. |
2451 With future applications in mind, we treat simultaneously the big category |
2451 With future applications in mind, we treat simultaneously the big $n{+}1$-category |
2452 of all $n$-categories and all sphere modules and also subcategories thereof. |
2452 of all $n$-categories and all sphere modules and also subcategories thereof. |
2453 When $n=1$ this is closely related to familiar $2$-categories consisting of |
2453 When $n=1$ this is closely related to the familiar $2$-category consisting of |
2454 algebras, bimodules and intertwiners (or a subcategory of that). |
2454 algebras, bimodules and intertwiners, or a subcategory of that. |
|
2455 (More generally, we can replace algebras with linear 1-categories.) |
|
2456 The ``bi" in ``bimodule" corresponds to the fact that a 0-sphere consists of two points. |
2455 The sphere module $n{+}1$-category is a natural generalization of the |
2457 The sphere module $n{+}1$-category is a natural generalization of the |
2456 algebra-bimodule-intertwiner 2-category to higher dimensions. |
2458 algebra-bimodule-intertwiner 2-category to higher dimensions. |
2457 |
2459 |
2458 Another possible name for this $n{+}1$-category is the $n{+}1$-category of defects. |
2460 Another possible name for this $n{+}1$-category is the $n{+}1$-category of defects. |
2459 The $n$-categories are thought of as representing field theories, and the |
2461 The $n$-categories are thought of as representing field theories, and the |
2461 In general, $m$-sphere modules are codimension $m{+}1$ defects; |
2463 In general, $m$-sphere modules are codimension $m{+}1$ defects; |
2462 the link of such a defect is an $m$-sphere decorated with defects of smaller codimension. |
2464 the link of such a defect is an $m$-sphere decorated with defects of smaller codimension. |
2463 |
2465 |
2464 \medskip |
2466 \medskip |
2465 |
2467 |
2466 While it is appropriate to call an $S^0$ module a bimodule, |
2468 %While it is appropriate to call an $S^0$ module a bimodule, |
2467 this is much less true for higher dimensional spheres, |
2469 %this is much less true for higher dimensional spheres, |
2468 so we prefer the term ``sphere module" for the general case. |
2470 %so we prefer the term ``sphere module" for the general case. |
2469 |
2471 |
2470 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
2472 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
2471 |
2473 |
2472 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
2474 The $1$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
2473 these first. |
2475 these first. |
2474 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
2476 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
2475 of $1$-category modules associated to decorated $n$-balls. |
2477 of $1$-category modules associated to decorated $n$-balls. |
2476 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
2478 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
2477 the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
2479 the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
2702 For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
2704 For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
2703 it could contain several. |
2705 it could contain several. |
2704 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
2706 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
2705 constructed out of labels taken from $L_j$ for $j<k$. |
2707 constructed out of labels taken from $L_j$ for $j<k$. |
2706 |
2708 |
2707 We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on |
2709 %We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on |
|
2710 We remind the reader again that $\cS$ depends on |
2708 the choice of $L_i$ above as well as the choice of |
2711 the choice of $L_i$ above as well as the choice of |
2709 families of inner products below. |
2712 families of inner products described below. |
2710 |
2713 |
2711 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
2714 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
2712 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
2715 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
2713 by elements of $L_j$. |
2716 by elements of $L_j$. |
2714 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
2717 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
2726 Next we define the $n{+}1$-morphisms of $\cS$. |
2729 Next we define the $n{+}1$-morphisms of $\cS$. |
2727 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
2730 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
2728 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2731 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2729 duality assumptions on the lower morphisms. |
2732 duality assumptions on the lower morphisms. |
2730 These are required because we define the spaces of $n{+}1$-morphisms by |
2733 These are required because we define the spaces of $n{+}1$-morphisms by |
2731 making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. |
2734 making arbitrary choices of incoming and outgoing boundaries for each $n{+}1$-ball. |
2732 The additional duality assumptions are needed to prove independence of our definition from these choices. |
2735 The additional duality assumptions are needed to prove independence of our definition from these choices. |
2733 |
2736 |
2734 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2737 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2735 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2738 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2736 Choose an $n{-}1$-sphere $E\sub \bd X$, transverse to $c$, which divides |
2739 Choose an $n{-}1$-sphere $E\sub \bd X$, transverse to $c$, which divides |
3245 \node at (-2,0) {$z \atop y$}; |
3248 \node at (-2,0) {$z \atop y$}; |
3246 \node at (6,0) {$1$}; |
3249 \node at (6,0) {$1$}; |
3247 \end{tikzpicture} |
3250 \end{tikzpicture} |
3248 $$ |
3251 $$ |
3249 |
3252 |
3250 \caption{intertwiners for a Morita equivalence}\label{morita-fig-2} |
3253 \caption{Intertwiners for a Morita equivalence}\label{morita-fig-2} |
3251 \end{figure} |
3254 \end{figure} |
3252 shows the intertwiners we need. |
3255 shows the intertwiners we need. |
3253 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3256 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3254 on the boundary. |
3257 on the boundary. |
3255 This is the 3-dimensional part of the data for the Morita equivalence. |
3258 This is the 3-dimensional part of the data for the Morita equivalence. |
3315 \end{tikzpicture} |
3318 \end{tikzpicture} |
3316 $$ |
3319 $$ |
3317 \caption{Identities for intertwiners}\label{morita-fig-3} |
3320 \caption{Identities for intertwiners}\label{morita-fig-3} |
3318 \end{figure} |
3321 \end{figure} |
3319 Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. |
3322 Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. |
3320 The modules corresponding leftmost and rightmost disks in the figure can be identified via the obvious isotopy. |
3323 The modules corresponding to the leftmost and rightmost disks in the figure can be identified via the obvious isotopy. |
3321 |
3324 |
3322 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
3325 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
3323 part of the Morita equivalence. |
3326 part of the Morita equivalence. |
3324 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
3327 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
3325 labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. |
3328 labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. |