diff -r 0c681fbb7b85 -r ff5483a2f789 text/ncat.tex --- a/text/ncat.tex Tue Jun 21 18:10:31 2011 -0700 +++ b/text/ncat.tex Wed Jun 22 11:06:33 2011 -0700 @@ -2593,26 +2593,27 @@ \medskip We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. -Recall that two 1-categories $C$ and $D$ are Morita equivalent if and only if they are equivalent +Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors. Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the $n{+}1$-category of sphere modules. Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in -dimensions 1 and $n+1$ (the middle dimensions come along for free), and this data must satisfy -identities corresponding to Morse cancellations in $n{+}1$-manifolds. +dimensions 1 and $n+1$ (the middle dimensions come along for free). +The $n{+}1$-dimensional part of the data must be invertible and satisfy +identities corresponding to Morse cancellations in $n$-manifolds. We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. -Let $C$ and $D$ be (unoriented) disklike 2-categories. +Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories. Let $\cS$ denote the 3-category of 2-category sphere modules. -The 1-dimensional part of the data for a Morita equivalence between $C$ and $D$ is a 0-sphere module $M = {}_CM_D$ -(categorified bimodule) connecting $C$ and $D$. +The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ +(categorified bimodule) connecting $\cC$ and $\cD$. Because of the full unoriented symmetry, this can also be thought of as a -0-sphere module ${}_DM_C$ connecting $D$ and $C$. +0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$. -We want $M$ to be an equivalence, so we need 2-morphisms in $\cS$ -between ${}_CM_D \otimes_D {}_DM_C$ and the identity 0-sphere module ${}_CC_C$, and similarly -with the roles of $C$ and $D$ reversed. +We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ +between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly +with the roles of $\cC$ and $\cD$ reversed. These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled cell complexes (cups and caps) in $B^2$ shown in Figure \nn{need figure}. @@ -2624,34 +2625,46 @@ 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. -\nn{?? note that, by symmetry, the x and y arrows of Fig xxxx are the same (up to rotation), as are the z and w arrows} +(Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{} are the same (up to rotation), as are the $h$ and $g$ arrows.) -In order for these 3-morphisms to be equivalences, they must satisfy identities corresponding to Morse cancellations -on 3-manifolds. +In order for these 3-morphisms to be equivalences, +they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition +they must satisfy identities corresponding to Morse cancellations on 2-manifolds. These are illustrated in Figure \nn{need figure}. Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. -For general $n$, we start with an $n$-category 0-sphere module $M$ which is the data for the 1-dimensional +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 $C$, $D$ and $M$; no additional data is needed for these parts. -The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must satisfy -identities corresponding to Morse cancellations in $n{+}1$-manifolds. +labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. +The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must +be invertible and satisfy +identities corresponding to Morse cancellations in $n$-manifolds. +\noop{ +One way of thinking of these conditions is as follows. +Given a decorated $n{+}1$-manifold, with a codimension 1 submanifold labeled by $\cM$ and +codimension 0 submanifolds labeled by $\cC$ and $\cD$, we can make any local modification we like without +changing +} +If $\cC$ and $\cD$ are Morita equivalent $n$-categories, then it is easy to show that for any $n-j$-manifold +$Y$ the $j$-categories $\cC(Y)$ and $\cD(Y)$ are Morita equivalent. +When $j=0$ this means that the TQFT Hilbert spaces $\cC(Y)$ and $\cD(Y)$ are isomorphic +(if we are enriching over vector spaces). \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1) -More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ -(categorified bimodule) connecting $C$ and $D$. -From $M$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = C$ or $D$. +More specifically, the 1-dimensional part of the data is a 0-sphere module $\cM = {}_\cCM_\cD$ +(categorified bimodule) connecting $\cC$ and $\cD$. +From $\cM$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = \cC$ or $\cD$. $N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$ (so the graph lives in $B^k\times I = B^{k+1}$). The positive side of the graph is labeled by $E$, the negative side by $E'$ -(where $C' = D$ and $D' = C$), and the codimension-1 -submanifold separating the positive and negative regions is labeled by $M$. +(where $\cC' = \cD$ and $\cD' = \cC$), and the codimension-1 +submanifold separating the positive and negative regions is labeled by $\cM$. We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting We plan on treating this in more detail in a future paper. \nn{should add a few more details}