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4 % '[handout]' for 4-up printed notes |
4 % '[handout]' for 4-up printed notes |
5 \documentclass[beamer]{beamer} |
5 \documentclass[beamer, compress]{beamer} |
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6 |
7 % change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes. |
7 % change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes. |
8 \newcommand{\pathtotrunk}{../../} |
8 \newcommand{\pathtotrunk}{../../} |
9 \input{\pathtotrunk talks/talk_preamble.tex} |
9 \input{\pathtotrunk talks/talk_preamble.tex} |
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37 \section{Overview} |
32 \section{Overview} |
38 |
33 |
39 \AtBeginSection[] |
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40 { |
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41 \begin{frame}<beamer> |
34 \begin{frame}<beamer> |
42 \frametitle{Outline} |
35 \frametitle{Blob homology} |
43 \tableofcontents[currentsection] |
36 \begin{quote} |
44 \end{frame} |
37 ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. |
45 } |
38 \end{quote} |
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39 \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}} |
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40 \tableofcontents |
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41 \end{frame} |
46 |
42 |
47 \begin{frame}{What is \emph{blob homology}?} |
43 \begin{frame}{What is \emph{blob homology}?} |
48 \begin{block}{} |
44 \begin{block}{} |
49 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. |
45 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. |
50 \end{block} |
46 \end{block} |
81 \begin{frame}{$n$-categories} |
77 \begin{frame}{$n$-categories} |
82 \begin{block}{Defining $n$-categories is fraught with difficulties} |
78 \begin{block}{Defining $n$-categories is fraught with difficulties} |
83 I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. |
79 I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. |
84 \end{block} |
80 \end{block} |
85 \begin{block}{} |
81 \begin{block}{} |
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82 Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. |
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83 \end{block} |
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84 |
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85 \begin{block}{} |
86 \begin{itemize} |
86 \begin{itemize} |
87 \item |
87 \item |
88 Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.\item |
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89 Defining $n$-categories: a choice of `shape' for morphisms. |
88 Defining $n$-categories: a choice of `shape' for morphisms. |
90 \item |
89 \item |
91 We allow all shapes! A vector space for every ball. |
90 We allow all shapes! A vector space for every ball. |
92 \item |
91 \item |
93 `Strong duality' is integral in our definition. |
92 `Strong duality' is integral in our definition. |
94 \end{itemize} |
93 \end{itemize} |
95 \end{block} |
94 \end{block} |
96 \end{frame} |
95 \end{frame} |
97 |
96 |
98 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
97 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
99 |
98 |
100 |
99 \section{Definition} |
101 \begin{frame}{Fields and pasting diagrams} |
100 \begin{frame}{Fields and pasting diagrams} |
102 \begin{block}{Pasting diagrams} |
101 \begin{block}{Pasting diagrams} |
103 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 $\cC$. |
102 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 $\cC$. |
104 \end{block} |
103 \end{block} |
105 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
104 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
106 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
105 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
107 \end{example} |
106 \end{example} |
108 \begin{block}{} |
107 \begin{block}{} |
109 Given a field on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
108 Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
110 \vspace{-3mm} |
109 \vspace{-3mm} |
111 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
112 \end{block} |
111 \end{block} |
113 \end{frame} |
112 \end{frame} |
114 |
113 |
117 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $A(\cM,; \cC)$. |
116 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $A(\cM,; \cC)$. |
118 \end{block} |
117 \end{block} |
119 |
118 |
120 \begin{block}{} |
119 \begin{block}{} |
121 \center |
120 \center |
122 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$. |
121 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
123 \end{block} |
122 \end{block} |
124 |
123 |
125 \begin{block}{} |
124 \begin{block}{} |
126 \vspace{-1mm} |
125 \vspace{-1mm} |
127 $$\bc_1(\cM; \cC) = \setc{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
126 $$\bc_1(\cM; \cC) = \setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
128 \end{block} |
127 \end{block} |
129 \vspace{-3.5mm} |
128 \vspace{-3.5mm} |
130 $$\mathfig{.5}{definition/single-blob}$$ |
129 $$\mathfig{.5}{definition/single-blob}$$ |
131 \vspace{-3mm} |
130 \vspace{-3mm} |
132 \begin{block}{} |
131 \begin{block}{} |
143 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
142 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
144 \end{block} |
143 \end{block} |
145 \begin{block}{} |
144 \begin{block}{} |
146 \vspace{-5mm} |
145 \vspace{-5mm} |
147 \begin{align*} |
146 \begin{align*} |
148 \bc_2^{\text{disjoint}} & = \roundframe{\mathfig{0.5}{definition/disjoint-blobs}} & u_i \in \ker{\text{ev}_{B_i}} |
147 \bc_2^{\text{disjoint}} & = \setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} |
149 \end{align*} |
148 \end{align*} |
150 \vspace{-4mm} |
149 \vspace{-4mm} |
151 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
150 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
152 \end{block} |
151 \end{block} |
153 \begin{block}{} |
152 \begin{block}{} |
154 \vspace{-5mm} |
153 \vspace{-5mm} |
155 \begin{align*} |
154 \begin{align*} |
156 \bc_2^{\text{nested}} & = \roundframe{\mathfig{0.5}{definition/nested-blobs}} & u \in \ker{\text{ev}_{B_1}} |
155 \bc_2^{\text{nested}} & = \setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0} |
157 \end{align*} |
156 \end{align*} |
158 \vspace{-4mm} |
157 \vspace{-4mm} |
159 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
158 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
160 \end{block} |
159 \end{block} |
161 \end{frame} |
160 \end{frame} |
162 |
161 |
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162 \begin{frame}{Definition, general case} |
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163 \begin{block}{} |
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164 $$\bc_k = \set{\mathfig{0.4}{tempkw/blobkdiagram}}$$ |
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165 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. |
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166 \end{block} |
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167 \begin{block}{} |
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168 \vspace{-2mm} |
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169 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
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170 \end{block} |
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171 \end{frame} |
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172 |
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173 \section{Properties} |
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174 \begin{frame}{An action of $\CH{\cM}$} |
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175 \begin{thm} |
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176 There's a chain map |
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177 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
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178 which is associative up to homotopy, and compatible with gluing. |
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179 \end{thm} |
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180 \begin{block}{} |
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181 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
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182 \end{block} |
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183 \end{frame} |
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184 |
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185 \begin{frame}{Gluing} |
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186 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
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187 \begin{itemize} |
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188 \item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
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189 \item[$m_k$:] reparametrising $[0,1]$ |
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190 \end{itemize} |
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191 \end{block} |
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192 \begin{block}{} |
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193 If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. |
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194 \end{block} |
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195 \begin{thm}[Gluing formula] |
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196 When $Y \sqcup Y^{\text{op}} \subset \bdy X$, |
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197 \vspace{-5mm} |
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198 \[ |
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199 \bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow. |
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200 \] |
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201 \end{thm} |
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202 In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. |
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203 \end{frame} |
163 \end{document} |
204 \end{document} |
164 % ---------------------------------------------------------------- |
205 % ---------------------------------------------------------------- |
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