44 \end{frame} |
44 \end{frame} |
45 } |
45 } |
46 |
46 |
47 \begin{frame}{What is \emph{blob homology}?} |
47 \begin{frame}{What is \emph{blob homology}?} |
48 \begin{block}{} |
48 \begin{block}{} |
49 The blob complex takes an $n$-manifold $M$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(M; \cC)$. |
49 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. |
50 \end{block} |
50 \end{block} |
51 \begin{tikzpicture} |
51 \tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm] |
52 \node (blobs) at (0,0) {$\bc_*(M; \cC)$}; |
52 \begin{tikzpicture}[] |
53 \node (skein) at (3,0) {$A(M; \cC)$}; |
53 \setbeamercovered{% |
54 \node (hoch) at (0,3) {$HH_*(\cA)$}; |
54 transparent=5, |
55 \path[->]<1-> (blobs) edge (skein); |
55 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, |
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56 again covered={\opaqueness<1->{50}} |
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57 } |
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58 |
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59 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
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60 \uncover<1>{ |
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61 \node[blue] (skein) at (4,0) {$A(\cM; \cC)$}; |
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62 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
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63 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
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64 } |
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65 |
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66 \uncover<2>{ |
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67 \node[blue] (hoch) at (0,3) {$HH_*(\cC)$}; |
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68 \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)}; |
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69 \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch); |
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70 } |
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71 |
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72 \uncover<3>{ |
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73 \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$}; |
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74 \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)}; |
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75 \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm); |
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76 } |
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77 |
56 \end{tikzpicture} |
78 \end{tikzpicture} |
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79 \end{frame} |
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80 |
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81 \begin{frame}{$n$-categories} |
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82 \begin{block}{Defining $n$-categories is fraught with difficulties} |
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83 I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. |
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84 \end{block} |
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85 \begin{block}{} |
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86 \begin{itemize} |
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87 \item |
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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. |
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90 \item |
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91 We allow all shapes! A vector space for every ball. |
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92 \item |
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93 `Strong duality' is integral in our definition. |
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94 \end{itemize} |
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95 \end{block} |
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96 \end{frame} |
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97 |
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98 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
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99 |
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100 |
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101 \begin{frame}{Fields and pasting diagrams} |
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102 \begin{block}{Pasting diagrams} |
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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 $\cC$. |
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104 \end{block} |
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105 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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106 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
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107 \end{example} |
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108 \begin{block}{} |
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109 Given a field on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
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110 \vspace{-3mm} |
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111 $$\text{ev}\Bigg(\roundframe{d \mathfig{0.12}{definition/evaluation1}} - \roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
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112 \end{block} |
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113 \end{frame} |
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114 |
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115 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
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116 \begin{block}{Motivation} |
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117 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $A(\cM,; \cC)$. |
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118 \end{block} |
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119 |
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120 \begin{block}{} |
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121 \center |
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122 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$. |
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123 \end{block} |
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124 |
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125 \begin{block}{} |
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126 \vspace{-1mm} |
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127 $$\bc_1(\cM; \cC) = \setc{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
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128 \end{block} |
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129 \vspace{-3.5mm} |
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130 $$\mathfig{.5}{definition/single-blob}$$ |
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131 \vspace{-3mm} |
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132 \begin{block}{} |
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133 \vspace{-6mm} |
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134 \begin{align*} |
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135 d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) |
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136 \end{align*} |
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137 \end{block} |
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138 \end{frame} |
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139 |
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140 \begin{frame}{Definition, $k=2$} |
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141 \begin{block}{} |
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142 \vspace{-1mm} |
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143 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
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144 \end{block} |
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145 \begin{block}{} |
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146 \vspace{-5mm} |
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147 \begin{align*} |
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148 \bc_2^{\text{disjoint}} & = \roundframe{\mathfig{0.5}{definition/disjoint-blobs}} & u_i \in \ker{\text{ev}_{B_i}} |
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149 \end{align*} |
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150 \vspace{-4mm} |
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151 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
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152 \end{block} |
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153 \begin{block}{} |
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154 \vspace{-5mm} |
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155 \begin{align*} |
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156 \bc_2^{\text{nested}} & = \roundframe{\mathfig{0.5}{definition/nested-blobs}} & u \in \ker{\text{ev}_{B_1}} |
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157 \end{align*} |
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158 \vspace{-4mm} |
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159 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
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160 \end{block} |
57 \end{frame} |
161 \end{frame} |
58 |
162 |
59 \end{document} |
163 \end{document} |
60 % ---------------------------------------------------------------- |
164 % ---------------------------------------------------------------- |
61 |
165 |