blob: comparison text/ncat.tex

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 \label{moddecss}
 Fix an ordinary $n$-category or $A_\infty$ $n$-category  $\cC$.
 Let $W$ be a $k$-manifold ($k\le n$),
 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
-and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
+and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each $Y_i$.
 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to
 the one appearing in \S \ref{ss:ncat_fields} above.
 (If $k = n$ and our $n$-categories are enriched, then
 $\cC(W, \cN)$ will have additional structure; see below.)
 \[
 	\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right)
 \]
 such that the restrictions to the various pieces of shared boundaries amongst the
 $X_a$ and $M_{ib}$ all agree.
-(That is, the fibered product over the boundary restriction maps.)
+%(That is, the fibered product over the boundary restriction maps.)
 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
 We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$.
 (As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever
 Modules are collections of functors together with some additional data, so we define morphisms
 of modules to be collections of natural transformations which are compatible with this
 additional data.
 More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors
-$\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets
+$\{\cX_k\}$ and $\{\cY_k\}$, for $1\le k\le n$, from marked $k$-balls to sets
 as in Module Axiom \ref{module-axiom-funct}.
 A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$
 satisfying:
 \begin{itemize}
 \item Each $g_k$ commutes with $\bd$.
 \label{ssec:spherecat}
 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules".
 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$,
 and the $n{+}1$-morphisms are intertwiners.
-With future applications in mind, we treat simultaneously the big category
+With future applications in mind, we treat simultaneously the big $n{+}1$-category
 of all $n$-categories and all sphere modules and also subcategories thereof.
-When $n=1$ this is closely related to familiar $2$-categories consisting of
+When $n=1$ this is closely related to the familiar $2$-category consisting of
-algebras, bimodules and intertwiners (or a subcategory of that).
+algebras, bimodules and intertwiners, or a subcategory of that.
+(More generally, we can replace algebras with linear 1-categories.)
+The ``bi" in ``bimodule" corresponds to the fact that a 0-sphere consists of two points.
 The sphere module $n{+}1$-category is a natural generalization of the
 algebra-bimodule-intertwiner 2-category to higher dimensions.
 Another possible name for this $n{+}1$-category is the $n{+}1$-category of defects.
 The $n$-categories are thought of as representing field theories, and the
 In general, $m$-sphere modules are codimension $m{+}1$ defects;
 the link of such a defect is an $m$-sphere decorated with defects of smaller codimension.
 \medskip
-While it is appropriate to call an $S^0$ module a bimodule,
+%While it is appropriate to call an $S^0$ module a bimodule,
-this is much less true for higher dimensional spheres,
+%this is much less true for higher dimensional spheres,
-so we prefer the term ``sphere module" for the general case.
+%so we prefer the term ``sphere module" for the general case.
 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces.
-The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe
+The $1$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe
 these first.
 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners
 of  $1$-category modules associated to decorated $n$-balls.
 We will see below that in order for these $n{+}1$-morphisms to satisfy all of
 the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume
 For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or
 it could contain several.
 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category
 constructed out of labels taken from $L_j$ for $j<k$.
-We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on
+%We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on
+We remind the reader again that $\cS$ depends on
 the choice of $L_i$ above as well as the choice of
-families of inner products below.
+families of inner products described below.
 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all
 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled
 by elements of $L_j$.
 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module
 Next we define the $n{+}1$-morphisms of $\cS$.
 The construction of the 0- through $n$-morphisms was easy and tautological, but the
 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional
 duality assumptions on the lower morphisms.
 These are required because we define the spaces of $n{+}1$-morphisms by
-making arbitrary choices of incoming and outgoing boundaries for each $n$-ball.
+making arbitrary choices of incoming and outgoing boundaries for each $n{+}1$-ball.
 The additional duality assumptions are needed to prove independence of our definition from these choices.
 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary
 by a cell complex labeled by 0- through $n$-morphisms, as above.
 Choose an $n{-}1$-sphere $E\sub \bd X$, transverse to $c$, which divides
 \node at (-2,0) {$z \atop y$};
 \node at (6,0) {$1$};
 \end{tikzpicture}
 $$
-\caption{intertwiners for a Morita equivalence}\label{morita-fig-2}
+\caption{Intertwiners for a Morita equivalence}\label{morita-fig-2}
 \end{figure}
 shows the intertwiners we need.
 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle
 on the boundary.
 This is the 3-dimensional part of the data for the Morita equivalence.
 \end{tikzpicture}
 $$
 \caption{Identities for intertwiners}\label{morita-fig-3}
 \end{figure}
 Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner.
-The modules corresponding leftmost and rightmost disks in the figure can be identified via the obvious isotopy.
+The modules corresponding to the leftmost and rightmost disks in the figure can be identified via the obvious isotopy.
 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional
 part of the Morita equivalence.
 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds
 labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts.

changeset 952	86389e393c17
parent 951	369f30add8d1
child 971	bbf14d934cb1