153
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1 |
% use options
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
2 |
% '[beamer]' for a digital projector
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
3 |
% '[trans]' for an overhead projector
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
4 |
% '[handout]' for 4-up printed notes
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
5 |
\documentclass[beamer]{beamer}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
6 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
7 |
% change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
8 |
\newcommand{\pathtotrunk}{../../}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
9 |
\input{\pathtotrunk talks/talk_preamble.tex}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
10 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
11 |
%\setbeameroption{previous slide on second screen=right}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
12 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
13 |
\author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker}
|
156
|
14 |
\institute{UC Berkeley / Miller Institute for Basic Research}
|
153
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
15 |
\title{Blob homology, part $\mathbb{I}$}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
16 |
\date{Homotopy Theory and Higher Algebraic Structures, UC Riverside, November 10 2009 \\ \url{http://tqft.net/UCR-blobs1}}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
17 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
18 |
\begin{document}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
19 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
20 |
\frame{\titlepage}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
21 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
22 |
\begin{frame}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
23 |
\frametitle{Outline}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
24 |
\tableofcontents
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
25 |
\end{frame}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
26 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
27 |
\beamertemplatetransparentcovered
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
28 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
29 |
\mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
30 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
31 |
\beamersetuncovermixins
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
32 |
{\opaqueness<1->{60}}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
33 |
{}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
34 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
35 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
36 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
37 |
\section{Overview}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
38 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
39 |
\AtBeginSection[]
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
40 |
{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
41 |
\begin{frame}<beamer>
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
42 |
\frametitle{Outline}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
43 |
\tableofcontents[currentsection]
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
44 |
\end{frame}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
45 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
46 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
47 |
\begin{frame}{What is \emph{blob homology}?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
48 |
\begin{block}{}
|
157
|
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}
|
|
51 |
\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]
|
|
52 |
\begin{tikzpicture}[]
|
|
53 |
\setbeamercovered{%
|
|
54 |
transparent=5,
|
|
55 |
% still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}},
|
|
56 |
again covered={\opaqueness<1->{50}}
|
|
57 |
}
|
|
58 |
|
|
59 |
\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};
|
|
60 |
\uncover<1>{
|
|
61 |
\node[blue] (skein) at (4,0) {$A(\cM; \cC)$};
|
|
62 |
\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};
|
|
63 |
\path[->](blobs) edge node[above] {$*= 0$} (skein);
|
|
64 |
}
|
|
65 |
|
|
66 |
\uncover<2>{
|
|
67 |
\node[blue] (hoch) at (0,3) {$HH_*(\cC)$};
|
|
68 |
\node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)};
|
|
69 |
\path[->](blobs) edge node[right] {$\cM = S^1$} (hoch);
|
|
70 |
}
|
|
71 |
|
|
72 |
\uncover<3>{
|
|
73 |
\node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$};
|
|
74 |
\node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)};
|
|
75 |
\path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm);
|
|
76 |
}
|
|
77 |
|
|
78 |
\end{tikzpicture}
|
|
79 |
\end{frame}
|
|
80 |
|
|
81 |
\begin{frame}{$n$-categories}
|
|
82 |
\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.
|
|
84 |
\end{block}
|
|
85 |
\begin{block}{}
|
|
86 |
\begin{itemize}
|
|
87 |
\item
|
|
88 |
Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.\item
|
|
89 |
Defining $n$-categories: a choice of `shape' for morphisms.
|
|
90 |
\item
|
|
91 |
We allow all shapes! A vector space for every ball.
|
|
92 |
\item
|
|
93 |
`Strong duality' is integral in our definition.
|
|
94 |
\end{itemize}
|
|
95 |
\end{block}
|
|
96 |
\end{frame}
|
|
97 |
|
|
98 |
\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}}
|
|
99 |
|
|
100 |
|
|
101 |
\begin{frame}{Fields and pasting diagrams}
|
|
102 |
\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 $\cC$.
|
153
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
104 |
\end{block}
|
157
|
105 |
\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category]
|
|
106 |
$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$
|
|
107 |
\end{example}
|
|
108 |
\begin{block}{}
|
|
109 |
Given a field on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}.
|
|
110 |
\vspace{-3mm}
|
|
111 |
$$\text{ev}\Bigg(\roundframe{d \mathfig{0.12}{definition/evaluation1}} - \roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$
|
|
112 |
\end{block}
|
|
113 |
\end{frame}
|
|
114 |
|
|
115 |
\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}
|
|
116 |
\begin{block}{Motivation}
|
|
117 |
A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $A(\cM,; \cC)$.
|
|
118 |
\end{block}
|
|
119 |
|
|
120 |
\begin{block}{}
|
|
121 |
\center
|
|
122 |
$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$.
|
|
123 |
\end{block}
|
|
124 |
|
|
125 |
\begin{block}{}
|
|
126 |
\vspace{-1mm}
|
|
127 |
$$\bc_1(\cM; \cC) = \setc{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$
|
|
128 |
\end{block}
|
|
129 |
\vspace{-3.5mm}
|
|
130 |
$$\mathfig{.5}{definition/single-blob}$$
|
|
131 |
\vspace{-3mm}
|
|
132 |
\begin{block}{}
|
|
133 |
\vspace{-6mm}
|
|
134 |
\begin{align*}
|
|
135 |
d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)
|
|
136 |
\end{align*}
|
|
137 |
\end{block}
|
|
138 |
\end{frame}
|
|
139 |
|
|
140 |
\begin{frame}{Definition, $k=2$}
|
|
141 |
\begin{block}{}
|
|
142 |
\vspace{-1mm}
|
|
143 |
$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$
|
|
144 |
\end{block}
|
|
145 |
\begin{block}{}
|
|
146 |
\vspace{-5mm}
|
|
147 |
\begin{align*}
|
|
148 |
\bc_2^{\text{disjoint}} & = \roundframe{\mathfig{0.5}{definition/disjoint-blobs}} & u_i \in \ker{\text{ev}_{B_i}}
|
|
149 |
\end{align*}
|
|
150 |
\vspace{-4mm}
|
|
151 |
$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$
|
|
152 |
\end{block}
|
|
153 |
\begin{block}{}
|
|
154 |
\vspace{-5mm}
|
|
155 |
\begin{align*}
|
|
156 |
\bc_2^{\text{nested}} & = \roundframe{\mathfig{0.5}{definition/nested-blobs}} & u \in \ker{\text{ev}_{B_1}}
|
|
157 |
\end{align*}
|
|
158 |
\vspace{-4mm}
|
|
159 |
$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$
|
|
160 |
\end{block}
|
153
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
161 |
\end{frame}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
162 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
163 |
\end{document}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
164 |
% ----------------------------------------------------------------
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
165 |
|