# HG changeset patch # User Kevin Walker # Date 1308845991 25200 # Node ID 233afb8cdae2035957b0754b2315047f315c68c6 # Parent 23675bc2a7688b542d7b4f16c32f590dce80d5f0# Parent dbe860c3d68617627042e1061962105c09192e57 Automated merge with https://tqft.net/hg/blob/ diff -r dbe860c3d686 -r 233afb8cdae2 pnas/PNAS-final.pdf Binary file pnas/PNAS-final.pdf has changed diff -r dbe860c3d686 -r 233afb8cdae2 text/intro.tex --- a/text/intro.tex Thu Jun 23 09:19:42 2011 -0700 +++ b/text/intro.tex Thu Jun 23 09:19:51 2011 -0700 @@ -80,7 +80,7 @@ In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category of sphere modules. -When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. +When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners. In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category (using a colimit along certain decompositions of a manifold into balls). @@ -96,6 +96,9 @@ The relationship between all these ideas is sketched in Figure \ref{fig:outline}. +% NB: the following tikz requires a *more recent* version of PGF than is distributed with MacTex 2010. +% grab the latest build from http://www.texample.net/tikz/builds/ +% unzip it in your personal tex tree, and run "mktexlsr ." there \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] \begin{figure}[t] diff -r dbe860c3d686 -r 233afb8cdae2 text/ncat.tex --- a/text/ncat.tex Thu Jun 23 09:19:42 2011 -0700 +++ b/text/ncat.tex Thu Jun 23 09:19:51 2011 -0700 @@ -489,7 +489,7 @@ \end{scope} \end{tikzpicture} $$ -\caption{Five examples of unions of pinched products}\label{pinched_prod_unions} +\caption{Six examples of unions of pinched products}\label{pinched_prod_unions} \end{figure} Note that $\bd X$ has a (possibly trivial) subdivision according to @@ -1507,7 +1507,7 @@ \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, -such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$. +such that $b_i$ and $b_{i+1}$ both map to (glue up to) $a_i$. \end{itemize} In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. The idea of the proof is to produce a similar zig-zag where everything antirefines to the same @@ -2079,13 +2079,13 @@ 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 intertwinors. +and the $n{+}1$-morphisms are intertwiners. With future applications in mind, we treat simultaneously the big 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 algebras, bimodules and intertwiners (or a subcategory of that). The sphere module $n{+}1$-category is a natural generalization of the -algebra-bimodule-intertwinor 2-category to higher dimensions. +algebra-bimodule-intertwiner 2-category to higher dimensions. Another possible name for this $n{+}1$-category is $n{+}1$-category of defects. The $n$-categories are thought of as representing field theories, and the @@ -2693,7 +2693,7 @@ We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. 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. +objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners. Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the $n{+}1$-category of sphere modules. @@ -2717,20 +2717,52 @@ cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. \begin{figure}[t] $$\mathfig{.65}{tempkw/morita1}$$ + + +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ + \draw[orange] (0,0) -- node[below] {$\cC$} (1,0); + \draw[blue] (1,0) -- node[below] {$\cD$} (2,0); + \draw[orange] (2,0) -- node[below] {$\cC$} (3,0); + \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; + \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; +}}; + +\node(R) at (6,0) {\tikz{ + \draw[orange] (0,0) -- node[below] {$\cC$} (3,0); + \node[label={\phantom{$\cM$}}] at (1.5,0) {}; +}}; + +\node at (-1,-1.5) { $\leftidx{_\cC}{(\cM \tensor_\cD \cM)}{_\cC}$ }; +\node at (7,-1.5) { $\leftidx{_\cC}{\cC}{_\cC}$ }; + +\draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{ + \draw (0,0) circle (16pt); +}}(R); + +\draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{ + \draw (0,0) circle (16pt); +}}(L); + + +\end{tikzpicture} +$$ + \caption{Cups and caps for free}\label{morita-fig-1} \end{figure} We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms between various compositions of these 2-morphisms and various identity 2-morphisms. -Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated +Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated to decorated circles. Figure \ref{morita-fig-2} \begin{figure}[t] $$\mathfig{.55}{tempkw/morita2}$$ -\caption{Intertwinors for a Morita equivalence}\label{morita-fig-2} +\caption{intertwiners for a Morita equivalence}\label{morita-fig-2} \end{figure} -shows the intertwinors we need. +shows the intertwiners we need. Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle on the boundary. This is the 3-dimensional part of the data for the Morita equivalence. @@ -2743,15 +2775,15 @@ These are illustrated in Figure \ref{morita-fig-3}. \begin{figure}[t] $$\mathfig{.65}{tempkw/morita3}$$ -\caption{Identities for intertwinors}\label{morita-fig-3} +\caption{Identities for intertwiners}\label{morita-fig-3} \end{figure} -Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. +Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. 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. -The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must +The $n{+}1$-dimensional part of the equivalence is given by certain intertwiners, and these intertwiners must be invertible and satisfy identities corresponding to Morse cancellations in $n$-manifolds.