# HG changeset patch # User Scott Morrison # Date 1277146636 14400 # Node ID 6876295aec26451ff74df09408d51c13f1d7d52f # Parent 6caac26b5c2991a14219c900231212580626a351 updating talk, more about n-categories diff -r 6caac26b5c29 -r 6876295aec26 diagrams/ncat/zz2.pdf Binary file diagrams/ncat/zz2.pdf has changed diff -r 6caac26b5c29 -r 6876295aec26 talks/20100625-StonyBrook/categorification.pdf Binary file talks/20100625-StonyBrook/categorification.pdf has changed diff -r 6caac26b5c29 -r 6876295aec26 talks/20100625-StonyBrook/categorification.tex --- a/talks/20100625-StonyBrook/categorification.tex Sun Jun 20 12:43:11 2010 -0700 +++ b/talks/20100625-StonyBrook/categorification.tex Mon Jun 21 14:57:16 2010 -0400 @@ -77,6 +77,86 @@ \end{tikzpicture} \end{frame} +\begin{frame}{$n$-categories} +\begin{block}{There are many definitions of $n$-categories!} +For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal $2$-categories. +\end{block} +\begin{block}{We have another definition!} +\emph{Many axioms}; geometric examples are easy, algebraic ones hard. +\begin{itemize} +%\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. +\item A vector space $\cC(B^n)$ for every $n$-ball $B$. +%\item From these, inductively +%\begin{itemize} +%\item define a set $\cC(S^k)$ for each $k$-sphere, $0 \leq k < n$, +%\item require a map $\cC(B^k) \to \cC(S^{k-1})$. +%\end{itemize} +\item An associative gluing map: with $B = \bigcup_i B_i$, balls glued together to form a ball, +$$\bigotimes \cC(B_i) \to \cC(B)$$ +(the $\tensor$ is fibered over `boundary restriction' maps). +\item ... +\end{itemize} +\end{block} +\end{frame} + +\begin{frame}{Cellulations of manifolds} +\begin{block}{} +Consider $\cell(M)$, the category of cellulations of a manifold $M$, with morphisms `antirefinements'. +\end{block} +\vspace{-4mm} +$$\mathfig{.35}{ncat/zz2}$$ +\vspace{-4mm} +\begin{block}{} +An $n$-category $\cC$ gives a functor from $\cell(M)$ to vector spaces. +\begin{description} +\item[objects] send a cellulation to the product of $\cC$ on each top-cell, restricting to the subset where boundaries agree +\item[morphisms] send an antirefinement to the appropriate gluing map. +\end{description} +\end{block} +\end{frame} + +\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} + +\section{Definition} +\begin{frame}{Fields} +\begin{block}{} +A field on $\cM^n$ is a choice of cellulation and a choice of $n$-morphism for each top-cell. +%$$\cF(\cM) = \bigoplus_{\cX \in \cell(M)} \bigotimes_{B \in \cX} \cC(B)$$ +\end{block} +\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] +$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ +\end{example} +\begin{block}{} +Given a field on a ball, we can evaluate it to a morphism using the gluing map. We call the kernel the \emph{null fields}. +\vspace{-3mm} +$$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ +\end{block} +\end{frame} + +\begin{frame}{Background: TQFT invariants} +\begin{defn} +We associate to an $n$-manifold $\cM$ the skein module +\vspace{-1mm} +$$\cA(\cM) = \cF(\cM) / \ker{ev},\vspace{-1mm}$$ +fields modulo fields which evaluate to zero inside some ball. +\end{defn} +Equivalently, $\cA(\cM)$ is the colimit of $\cC$ along $\cell(M)$. + +\vspace{4mm} +%\begin{itemize} +%\item We can also associate a $k$-category to an $n-k$-manifold. +%\item We don't assign a number to an $n+1$-manifold (a `decapitated' extended TQFT). +%\end{itemize} +$\cA(Y \times [0,1])$ is a $1$-category, and when $Y \subset \bdy X$, $\cA(X)$ is a module over $\cA(Y \times [0,1])$. +\begin{thm}[Gluing formula] +When $Y \sqcup Y^{\text{op}} \subset \bdy X$, +\vspace{-1mm} +\[ + \cA(X \bigcup_Y \selfarrow) \iso \cA(X) \bigotimes_{\cA(Y \times [0,1])} \selfarrow. +\] +\end{thm} +\end{frame} + \begin{frame}{Motivation: Khovanov homology as a $4$d TQFT} \begin{thm} Khovanov homology gives a $4$-category: @@ -93,7 +173,7 @@ \begin{frame}{Computations are hard} \begin{block}{} -The corresponding $4$-manifold invariant is hard to compute, because the TQFT skein module construction breaks the exact triangle for resolving a crossing. +This invariant is hard to compute, because the TQFT skein module construction breaks the exact triangle for resolving a crossing. \vspace{-0.3cm} \begin{align*} \begin{tikzpicture} @@ -148,76 +228,16 @@ \end{conj} \end{frame} -\begin{frame}{$n$-categories} -\begin{block}{Defining $n$-categories is fraught with difficulties} -For now, I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. -\end{block} -\begin{block}{} -Later, I'll explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. -\end{block} - -\begin{block}{} -\begin{itemize} -\item -Defining $n$-categories: a choice of `shape' for morphisms. -\item -We allow all shapes! A vector space for every ball. -\item -`Strong duality' is integral in our definition. -\end{itemize} -\end{block} -\end{frame} - -\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} - -\section{Definition} -\begin{frame}{Fields and pasting diagrams} -\begin{block}{Pasting diagrams} -Fix an $n$-category with strong duality $\cC$. A \emph{field} on $\cM$ is a pasting diagram drawn on $\cM$, with cells labelled by morphisms from $\cC$. -\end{block} -\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] -$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ -\end{example} -\begin{block}{} -Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. -\vspace{-3mm} -$$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ -\end{block} -\end{frame} - -\begin{frame}{Background: TQFT invariants} -\begin{defn} -A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. -\end{defn} -(`decapitated': no numerical invariants of $n+1$-manifolds.) - -\begin{block}{} -If the manifold has boundary, we get a category. Objects are boundary data, $\Hom{\cA(\cM)}{x}{y} = \cA(\cM; x,y)$. -\end{block} - -\begin{block}{} -We want to extend `all the way down'. The $k$-category associated to the $n-k$-manifold $\cY$ is $\cA(\cY \times B^k)$. -\end{block} - -\begin{defn} -Given an $n$-category $\cC$, the associated TQFT is -\vspace{-3mm} -$$\cA(\cM) = \cF(M) / \ker{ev},$$ - -\vspace{-3mm} -fields modulo fields which evaluate to zero inside some ball. -\end{defn} -\end{frame} \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} \begin{block}{Motivation} -A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$. +A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM; \cC)$. \end{block} \mode{\vspace{-5mm}} \begin{block}{} \center -$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. +$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$. \end{block} \begin{block}{} @@ -264,7 +284,7 @@ \begin{frame}{Definition, general case} \begin{block}{} $$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$ -$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. +$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by fields that evaluate to zero. \end{block} \begin{block}{} \vspace{-2mm} @@ -306,12 +326,23 @@ \end{block} \end{frame} +\mode{ +\begin{frame}{An action of $\CH{\cM}$} +\begin{proof} +\begin{description} +\item[Step 1] If $\cM=B^n$ or a union of balls, there's a unique chain map, since $\bc_*(B^n; \cC) \htpy \cC$ is concentrated in homological degree $0$. +\item[Step 2] Fix an open cover $\cU$ of balls. \\ A family of homeomorphisms $P^k \to \Homeo(\cM)$ can be broken up in into pieces, each of which is supported in at most $k$ open sets from $\cU$. \qedhere +\end{description} +\end{proof} +\end{frame} +} + \begin{frame}{Gluing} \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} -\begin{itemize} -\item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ -\item[$m_k$:] reparametrising $[0,1]$ -\end{itemize} +\begin{description} +\item[multiplication ($m_2$):] gluing $[0,1] \simeq [0,1] \cup [0,1]$ +\item[associativity up to homotopy ($m_k$):] reparametrising $[0,1]$ using the action of $\CH{[0,1]}$. +\end{description} \end{block} \begin{block}{} If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. @@ -344,14 +375,14 @@ \begin{frame}{Maps to a space} \begin{block}{} -Fix a target space $T$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(T)$ defined by -$$\pi_{\leq n}^\infty(T)(B) = C_*(\Maps(B\to T)).$$ +Fix a target space $\cT$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(\cT)$ defined by +$$\pi_{\leq n}^\infty(\cT)(B) = C_*(\Maps(B\to \cT)).$$ \end{block} \begin{thm} The blob complex recovers mapping spaces: -$$\bc_*(M; \pi_{\leq n}^\infty(T)) \iso C_*(\Maps(M \to T))$$ +$$\bc_*(\cM; \pi_{\leq n}^\infty(\cT)) \iso C_*(\Maps(\cM \to \cT))$$ \end{thm} -This generalizes a result of Lurie: if $T$ is $n-1$ connected, $\pi_{\leq n}^\infty(T)$ is an $E_n$-algebra and the blob complex is the same as his topological chiral homology. +This generalizes a result of Lurie: if $\cT$ is $n-1$ connected, $\pi_{\leq n}^\infty(\cT)$ is an $E_n$-algebra and the blob complex is the same as his topological chiral homology. \end{frame} \end{document} diff -r 6caac26b5c29 -r 6876295aec26 talks/20100625-StonyBrook/handout.pdf Binary file talks/20100625-StonyBrook/handout.pdf has changed