# HG changeset patch # User Scott Morrison # Date 1277062991 25200 # Node ID 6caac26b5c2991a14219c900231212580626a351 # Parent e5867a64cae5ee3679144a35eb72bd5def9e6ab3 some updates for a longer talk diff -r e5867a64cae5 -r 6caac26b5c29 talks/20100625-StonyBrook/categorification.pdf Binary file talks/20100625-StonyBrook/categorification.pdf has changed diff -r e5867a64cae5 -r 6caac26b5c29 talks/20100625-StonyBrook/categorification.tex --- a/talks/20100625-StonyBrook/categorification.tex Sun Jun 20 11:00:57 2010 -0700 +++ b/talks/20100625-StonyBrook/categorification.tex Sun Jun 20 12:43:11 2010 -0700 @@ -12,8 +12,8 @@ \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}} +\title{The blob complex} +\date{Low-Dimensional Topology and Categorification, \\Stony Brook University, June 21-25 2010 \\ \begin{description}\item[slides:]\url{http://tqft.net/sunysb-blobs} \item[paper:]\url{http://tqft.net/blobs}\end{description}} \begin{document} @@ -21,6 +21,9 @@ \beamertemplatetransparentcovered +\setbeamertemplate{navigation symbols}{} % no navigation symbols, please + + \mode{\setbeamercolor{block title}{bg=green!40!black}} \beamersetuncovermixins @@ -32,7 +35,7 @@ \section{Overview} \begin{frame} - \frametitle{Blob homology} + \frametitle{The blob complex} \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} @@ -40,7 +43,7 @@ \tableofcontents \end{frame} -\begin{frame}{What is \emph{blob homology}?} +\begin{frame}{What is \emph{the blob complex}?} \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} @@ -53,19 +56,19 @@ } \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; -\uncover<1>{ +\uncover<2>{ \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>{ +\uncover<3>{ \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>{ +\uncover<4>{ \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); @@ -74,12 +77,83 @@ \end{tikzpicture} \end{frame} +\begin{frame}{Motivation: Khovanov homology as a $4$d TQFT} +\begin{thm} +Khovanov homology gives a $4$-category: +\begin{description} +\item[3-morphisms] tangles, with the usual $3$ operations, +\item[4-morphisms] $\Hom{Kh}{T_1}{T_2} = Kh(T_1 \cup \bar{T_2})$, composition defined by saddle cobordisms +\end{description} +\end{thm} +\begin{block}{} +There is a corresponding $4$-manifold invariant. Given $L \subset \bdy W^4$, it associates a doubly-graded vector space $\cA(W, L; Kh)$. +$$\cA(B^4, L; Kh) \iso Kh(L)$$ +\end{block} +\end{frame} + +\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. +\vspace{-0.3cm} +\begin{align*} +\begin{tikzpicture} +\node(a) at (0,0) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] +\node[outer sep=-1pt] (x) at (0,0){}; + \draw (x.45)-- (.5,.5); + \draw (x.135) -- (-.5,.5); + \draw (x.315) -- (.5,-.5); + \draw (x.45) -- (-.5,-.5); +\end{tikzpicture}\right)$}; +\node(b) at (-1.2,-1.5) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] + \draw (1.5,.5) .. controls (2,0) .. (1.5,-.5); + \draw (2.5,.5) .. controls (2,0) .. (2.5,-.5); +\end{tikzpicture}\right)$}; +\node(c) at (1.2,-1.5) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] + \draw (3.5,.5) .. controls (4,0) .. (4.5,.5); + \draw (3.5,-.5) .. controls (4,0) .. (4.5,-.5); +\end{tikzpicture}\right)$}; +\draw[->] (a) -- (b); +\draw[->] (b) -- (c); +\draw[->] (c) -- (a); +\end{tikzpicture} +\qquad \qquad +\begin{tikzpicture} +\node(a) at (0,0) {$\cA\left(M, \begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] +\node[outer sep=-1pt] (x) at (0,0){}; + \draw (x.45)-- (.5,.5); + \draw (x.135) -- (-.5,.5); + \draw (x.315) -- (.5,-.5); + \draw (x.45) -- (-.5,-.5); +\end{tikzpicture}\right)$}; +\node(b) at (-1.4,-1.5) {$\cA\left(M, \begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] + \draw (1.5,.5) .. controls (2,0) .. (1.5,-.5); + \draw (2.5,.5) .. controls (2,0) .. (2.5,-.5); +\end{tikzpicture}\right)$}; +\node(c) at (1.4,-1.5) {$\cA\left(M,\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] + \draw (3.5,.5) .. controls (4,0) .. (4.5,.5); + \draw (3.5,-.5) .. controls (4,0) .. (4.5,-.5); +\end{tikzpicture}\right)$}; +\node at (0,-0.75) {\Large \color{red} ?}; +\draw[dashed] (a) -- (b); +\draw[dashed] (b) -- (c); +\draw[dashed] (c) -- (a); +\end{tikzpicture} +\end{align*}\vspace{-1cm} +\end{block} +There is a spectral sequence converging to $0$ relating the blob homologies for the triangle of resolutions. +\begin{conj} +It may be possible to compute the skein module +%$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ +by first computing the entire blob homology. +\end{conj} +\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. +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}{} -Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. +Later, I'll explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. \end{block} \begin{block}{} @@ -102,7 +176,7 @@ 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] -$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ +$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ \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}. @@ -140,6 +214,7 @@ 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 $\cM$. @@ -153,6 +228,7 @@ $$\mathfig{.5}{definition/single-blob}$$ \vspace{-3mm} \begin{block}{} +\mode{\vspace{-5mm}} \vspace{-6mm} \begin{align*} d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) @@ -163,9 +239,11 @@ \begin{frame}{Definition, $k=2$} \begin{block}{} \vspace{-1mm} +\mode{\vspace{-5mm}} $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ \end{block} \begin{block}{} +\mode{\vspace{-5mm}} \vspace{-5mm} \begin{align*} \bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} @@ -200,6 +278,7 @@ \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} +\mode{\vspace{-3mm}} \begin{block}{} The Hochschild complex is `coinvariants of the bar resolution' \vspace{-2mm} @@ -227,22 +306,6 @@ \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} @@ -262,6 +325,35 @@ \end{thm} In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. \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 $\Hom{\bc_*(\bdy M)}{\bc_*(\cM)}{\bc_*(\cM)}$ `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}{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 +$$\pi_{\leq n}^\infty(T)(B) = C_*(\Maps(B\to T)).$$ +\end{block} +\begin{thm} +The blob complex recovers mapping spaces: +$$\bc_*(M; \pi_{\leq n}^\infty(T)) \iso C_*(\Maps(M \to T))$$ +\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. +\end{frame} + \end{document} % ---------------------------------------------------------------- diff -r e5867a64cae5 -r 6caac26b5c29 talks/20100625-StonyBrook/handout.pdf Binary file talks/20100625-StonyBrook/handout.pdf has changed