diff -r 43033c337dfa -r e5867a64cae5 talks/20100625-StonyBrook/categorification.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/talks/20100625-StonyBrook/categorification.tex Sun Jun 20 11:00:57 2010 -0700 @@ -0,0 +1,267 @@ +% use options +% '[beamer]' for a digital projector +% '[trans]' for an overhead projector +% '[handout]' for 4-up printed notes +\documentclass[beamer, compress]{beamer} + +% change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes. +\newcommand{\pathtotrunk}{../../} +\input{\pathtotrunk talks/talk_preamble.tex} + +%\setbeameroption{previous slide on second screen=right} + +\author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker} +\institute{UC Berkeley / Miller Institute for Basic Research} +\title{Blob homology, part $\mathbb{I}$} +\date{Homotopy Theory and Higher Algebraic Structures, UC Riverside, November 10 2009 \\ \begin{description}\item[slides, part $\mathbb{I}$:]\url{http://tqft.net/UCR-blobs1} \item[slides, part $\mathbb{II}$:]\url{http://tqft.net/UCR-blobs2} \item[draft:]\url{http://tqft.net/blobs}\end{description}} + +\begin{document} + +\frame{\titlepage} + +\beamertemplatetransparentcovered + +\mode{\setbeamercolor{block title}{bg=green!40!black}} + +\beamersetuncovermixins +{\opaqueness<1->{60}} +{} + + + +\section{Overview} + + \begin{frame} + \frametitle{Blob homology} + \begin{quote} + ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. + \end{quote} + \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}} + \tableofcontents +\end{frame} + +\begin{frame}{What is \emph{blob homology}?} +\begin{block}{} +The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. +\end{block} +\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm] +\begin{tikzpicture}[] +\setbeamercovered{% + transparent=5, +% still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, + again covered={\opaqueness<1->{50}} +} + +\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; +\uncover<1>{ +\node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; +\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; +\path[->](blobs) edge node[above] {$*= 0$} (skein); +} + +\uncover<2>{ + \node[blue] (hoch) at (0,3) {$HH_*(\cC)$}; + \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)}; + \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch); +} + +\uncover<3>{ + \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$}; + \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)}; + \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm); +} + +\end{tikzpicture} +\end{frame} + +\begin{frame}{$n$-categories} +\begin{block}{Defining $n$-categories is fraught with difficulties} +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}{} +Kevin's talk (part $\mathbb{II}$) will 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] +$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ +\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 $\cA(\cM,; \cC)$. +\end{block} + +\begin{block}{} +\center +$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. +\end{block} + +\begin{block}{} +\vspace{-1mm} +$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ +\end{block} +\vspace{-3.5mm} +$$\mathfig{.5}{definition/single-blob}$$ +\vspace{-3mm} +\begin{block}{} +\vspace{-6mm} +\begin{align*} +d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) +\end{align*} +\end{block} +\end{frame} + +\begin{frame}{Definition, $k=2$} +\begin{block}{} +\vspace{-1mm} +$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ +\end{block} +\begin{block}{} +\vspace{-5mm} +\begin{align*} +\bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} +\end{align*} +\vspace{-4mm} +$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ +\end{block} +\begin{block}{} +\vspace{-5mm} +\begin{align*} +\bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0} +\end{align*} +\vspace{-4mm} +$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ +\end{block} +\end{frame} + +\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. +\end{block} +\begin{block}{} +\vspace{-2mm} +$$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ +\end{block} +\end{frame} + +\section{Properties} +\begin{frame}{Hochschild homology} +\begin{block}{TQFT on $S^1$ is `coinvariants'} +\vspace{-3mm} +$$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$ +\end{block} +\begin{block}{} +The Hochschild complex is `coinvariants of the bar resolution' +\vspace{-2mm} +$$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ +\end{block} +\begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$] +$$m \tensor a \mapsto +\roundframe{\mathfig{0.35}{hochschild/1-chains}} +$$ +\vspace{-5mm} +\begin{align*} +u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 &= \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} +\end{align*} +\end{thm} +\end{frame} + +\begin{frame}{An action of $\CH{\cM}$} +\begin{thm} +There's a chain map +$$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ +which is associative up to homotopy, and compatible with gluing. +\end{thm} +\begin{block}{} +Taking $H_0$, this is the mapping class group acting on a TQFT skein module. +\end{block} +\end{frame} + +\begin{frame}{Higher Deligne conjecture} +\begin{block}{Deligne conjecture} +Chains on the little discs operad acts on Hochschild cohomology. +\end{block} + +\begin{block}{} +Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'. +\end{block} + +\begin{block}{Theorem* (Higher Deligne conjecture)} +\scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.} +\vspace{-3mm} +$$\mathfig{.85}{deligne/manifolds}$$ +\end{block} +\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} +\end{block} +\begin{block}{} +If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. +\end{block} +\begin{thm}[Gluing formula] +When $Y \sqcup Y^{\text{op}} \subset \bdy X$, +\vspace{-5mm} +\[ + \bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow. +\] +\end{thm} +In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. +\end{frame} +\end{document} +% ---------------------------------------------------------------- +