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13 \author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker} |
13 \author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker} |
14 \institute{UC Berkeley / Miller Institute for Basic Research} |
14 \institute{UC Berkeley / Miller Institute for Basic Research} |
15 \title{The blob complex} |
15 \title{The blob complex} |
16 \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}} |
16 \date{ |
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17 Low-Dimensional Topology and Categorification, \\ |
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18 Stony Brook University, June 21-25 2010 \\ |
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19 \begin{description} |
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20 \item[slides:]\url{http://tqft.net/talks} |
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21 \item[paper:]\url{http://tqft.net/blobs} |
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22 % \item[shameless plug:]\url{http://mathoverflow.net} |
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23 \end{description} |
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24 } |
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25 |
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26 \listfiles |
17 |
27 |
18 \begin{document} |
28 \begin{document} |
19 |
29 |
20 \frame{\titlepage} |
30 \frame{\titlepage} |
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30 {\opaqueness<1->{60}} |
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31 {} |
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32 |
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33 |
31 |
34 |
32 |
35 \section{Overview} |
33 \section{Overview} |
36 |
34 |
37 \begin{frame}<beamer> |
35 \begin{frame}<beamer> |
75 } |
73 } |
76 |
74 |
77 \end{tikzpicture} |
75 \end{tikzpicture} |
78 \end{frame} |
76 \end{frame} |
79 |
77 |
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78 \section{TQFTs} |
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79 |
80 \begin{frame}{$n$-categories} |
80 \begin{frame}{$n$-categories} |
81 \begin{block}{There are many definitions of $n$-categories!} |
81 \begin{block}{There are many definitions of $n$-categories!} |
82 For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal $2$-categories. |
82 For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal $2$-categories. |
83 \end{block} |
83 \end{block} |
84 \begin{block}{We have another definition!} |
84 \begin{block}{We have another definition: \emph{topological $n$-categories}} |
85 \emph{Many axioms}; geometric examples are easy, algebraic ones hard. |
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86 \begin{itemize} |
85 \begin{itemize} |
87 %\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. |
86 %\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. |
88 \item A vector space $\cC(B^n)$ for every $n$-ball $B$. |
87 \item A vector space $\cC(B^n)$ for every $n$-ball $B$. |
89 %\item From these, inductively |
88 %\item From these, inductively |
90 %\begin{itemize} |
89 %\begin{itemize} |
95 $$\bigotimes \cC(B_i) \to \cC(B)$$ |
94 $$\bigotimes \cC(B_i) \to \cC(B)$$ |
96 (the $\tensor$ is fibered over `boundary restriction' maps). |
95 (the $\tensor$ is fibered over `boundary restriction' maps). |
97 \item ... |
96 \item ... |
98 \end{itemize} |
97 \end{itemize} |
99 \end{block} |
98 \end{block} |
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99 These are easy to check for geometric examples, hard to check for algebraic examples. |
100 \end{frame} |
100 \end{frame} |
101 |
101 |
102 \begin{frame}{Cellulations of manifolds} |
102 \begin{frame}{Cellulations of manifolds} |
103 \begin{block}{} |
103 \begin{block}{} |
104 Consider $\cell(M)$, the category of cellulations of a manifold $M$, with morphisms `antirefinements'. |
104 Consider $\cell(M)$, the category of cellulations of a manifold $M$, with morphisms `antirefinements'. |
115 \end{block} |
115 \end{block} |
116 \end{frame} |
116 \end{frame} |
117 |
117 |
118 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
118 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
119 |
119 |
120 \section{Definition} |
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121 \begin{frame}{Fields} |
120 \begin{frame}{Fields} |
122 \begin{block}{} |
121 \begin{block}{} |
123 A field on $\cM^n$ is a choice of cellulation and a choice of $n$-morphism for each top-cell. |
122 A field on $\cM^n$ is a choice of cellulation and a choice of $n$-morphism for each top-cell (with matching boundaries). |
124 %$$\cF(\cM) = \bigoplus_{\cX \in \cell(M)} \bigotimes_{B \in \cX} \cC(B)$$ |
123 %$$\cF(\cM) = \bigoplus_{\cX \in \cell(M)} \bigotimes_{B \in \cX} \cC(B)$$ |
125 \end{block} |
124 \end{block} |
126 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
125 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
127 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ |
126 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ |
128 \end{example} |
127 \end{example} |
226 %$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ |
225 %$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ |
227 by first computing the entire blob homology. |
226 by first computing the entire blob homology. |
228 \end{conj} |
227 \end{conj} |
229 \end{frame} |
228 \end{frame} |
230 |
229 |
231 |
230 \section{Definition} |
232 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
231 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
233 \begin{block}{Motivation} |
232 \begin{block}{Motivation} |
234 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM; \cC)$. |
233 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM; \cC)$. |
235 \end{block} |
234 \end{block} |
236 |
235 |
294 |
293 |
295 \section{Properties} |
294 \section{Properties} |
296 \begin{frame}{Hochschild homology} |
295 \begin{frame}{Hochschild homology} |
297 \begin{block}{TQFT on $S^1$ is `coinvariants'} |
296 \begin{block}{TQFT on $S^1$ is `coinvariants'} |
298 \vspace{-3mm} |
297 \vspace{-3mm} |
299 $$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$ |
298 $$\cA(S^1, A) = \Complex\set{\roundframe{ |
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299 \tikz{\draw (0,0) circle (0.4); \foreach \q/\l in {90/a, 210/b, 330/c} {\draw[fill=red] (\q:0.4) circle (0.075); \node at (\q:0.6) {\l};}} |
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300 }} |
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301 \scalebox{2}{$/$} |
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302 \set{\roundframe{\tikz{\draw (-30:0.4) arc (-30:210:0.4); \draw[fill=red] (90:0.4) circle (0.075); \node at (90:0.65) {$ab$};}} - \roundframe{ |
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303 \tikz{\draw (-30:0.4) arc (-30:210:0.4); \foreach \q/\l in {120/a, 60/b} {\draw[fill=red] (\q:0.4) circle (0.075); \node at (\q:0.65) {\l};}}}} = A/(ab-ba)$$ |
300 \end{block} |
304 \end{block} |
301 \mode<handout>{\vspace{-3mm}} |
305 \mode<handout>{\vspace{-3mm}} |
302 \begin{block}{} |
306 \begin{block}{Blob homology on $S^1$ is Hochschild homology} |
303 The Hochschild complex is `coinvariants of the bar resolution' |
307 The Hochschild complex is `coinvariants of the bar resolution' |
304 \vspace{-2mm} |
308 \vspace{-2mm} |
305 $$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ |
309 $$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ |
306 \end{block} |
310 |
307 \begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$] |
311 We check universal properties, as it's hard to directly construct an isomorphism. |
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312 \noop{ |
308 $$m \tensor a \mapsto |
313 $$m \tensor a \mapsto |
309 \roundframe{\mathfig{0.35}{hochschild/1-chains}} |
314 \roundframe{\mathfig{0.35}{hochschild/1-chains}} |
310 $$ |
315 $$ |
311 \vspace{-5mm} |
316 \vspace{-5mm} |
312 \begin{align*} |
317 \begin{align*} |
313 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} |
318 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} |
314 \end{align*} |
319 \end{align*} |
315 \end{thm} |
320 } |
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321 \end{block} |
316 \end{frame} |
322 \end{frame} |
317 |
323 |
318 \begin{frame}{An action of $\CH{\cM}$} |
324 \begin{frame}{An action of $\CH{\cM}$} |
319 \begin{thm} |
325 \begin{thm} |
320 There's a chain map |
326 There's a chain map |
321 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
327 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
322 which is associative up to homotopy, and compatible with gluing. |
328 which is associative up to homotopy, and compatible with gluing. |
323 \end{thm} |
329 \end{thm} |
324 \begin{block}{} |
330 \begin{block}{} |
325 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
331 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
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332 $$H_0(\Homeo(\cM)) \tensor \cA(\cM) \to \cA(\cM).$$ |
326 \end{block} |
333 \end{block} |
327 \end{frame} |
334 \end{frame} |
328 |
335 |
329 \mode<beamer>{ |
336 \mode<beamer>{ |
330 \begin{frame}{An action of $\CH{\cM}$} |
337 \begin{frame}{An action of $\CH{\cM}$} |
375 |
382 |
376 \begin{frame}{Maps to a space} |
383 \begin{frame}{Maps to a space} |
377 \begin{block}{} |
384 \begin{block}{} |
378 Fix a target space $\cT$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(\cT)$ defined by |
385 Fix a target space $\cT$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(\cT)$ defined by |
379 $$\pi_{\leq n}^\infty(\cT)(B) = C_*(\Maps(B\to \cT)).$$ |
386 $$\pi_{\leq n}^\infty(\cT)(B) = C_*(\Maps(B\to \cT)).$$ |
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387 (Here $B$ is an $n$-ball.) |
380 \end{block} |
388 \end{block} |
381 \begin{thm} |
389 \begin{thm} |
382 The blob complex recovers mapping spaces: |
390 The blob complex recovers mapping spaces: |
383 $$\bc_*(\cM; \pi_{\leq n}^\infty(\cT)) \iso C_*(\Maps(\cM \to \cT))$$ |
391 $$\bc_*(\cM; \pi_{\leq n}^\infty(\cT)) \iso C_*(\Maps(\cM \to \cT))$$ |
384 \end{thm} |
392 \end{thm} |
385 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. |
393 This generalizes a result of Lurie: if $\cT$ is $n-1$ connected, $\pi_{\leq n}^\infty(\cT)$ is an $E_n$-algebra and in this special case the blob complex is presumably the same as his topological chiral homology. |
386 \end{frame} |
394 \end{frame} |
387 |
395 |
388 \end{document} |
396 \end{document} |
389 % ---------------------------------------------------------------- |
397 % ---------------------------------------------------------------- |
390 |
398 |