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\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 \\ \url{http://tqft.net/UCR-blobs1}}

\begin{document}

\frame{\titlepage}

\begin{frame}

       \frametitle{Outline}

       \tableofcontents

\end{frame}

\beamertemplatetransparentcovered 

\mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}}

\beamersetuncovermixins 

{\opaqueness<1->{60}} 

{} 

\section{Overview}

\AtBeginSection[]

{

   \begin{frame}<beamer>

       \frametitle{Outline}

       \tableofcontents[currentsection]

   \end{frame}

}

\begin{frame}{What is \emph{blob homology}?}

\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}

\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]

\begin{tikzpicture}[]

\setbeamercovered{%

 transparent=5,

% still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}},

 again covered={\opaqueness<1->{50}}

}

\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};

\uncover<1>{

\node[blue] (skein) at (4,0) {$A(\cM; \cC)$};

\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};

\path[->](blobs) edge node[above] {$*= 0$} (skein);

}

\uncover<2>{

  \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>{

  \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);

}

\end{tikzpicture}

\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.

\end{block}

\begin{block}{}

\begin{itemize}

\item

Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.\item

Defining $n$-categories: a choice of `shape' for morphisms.

\item

We allow all shapes! A vector space for every ball.

\item

`Strong duality' is integral in our definition.

\end{itemize}

\end{block}

\end{frame}

\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}}

\begin{frame}{Fields and pasting diagrams}

\begin{block}{Pasting diagrams}

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]

$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$

\end{example}

\begin{block}{}

Given a field on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}.

\vspace{-3mm}

\end{block}

\end{frame}

\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}

\begin{block}{Motivation}

A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $A(\cM,; \cC)$.

\end{block}

\begin{block}{}

\center

$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$.

\end{block}

\begin{block}{}

\vspace{-1mm}

$$\bc_1(\cM; \cC) = \setc{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$

\end{block}

\vspace{-3.5mm}

$$\mathfig{.5}{definition/single-blob}$$

\vspace{-3mm}

\begin{block}{}

\vspace{-6mm}

\begin{align*}

d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)

\end{align*}

\end{block}

\end{frame}

\begin{frame}{Definition, $k=2$}

\begin{block}{}

\vspace{-1mm}

$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$

\end{block}

\begin{block}{}

\vspace{-5mm}

\begin{align*}

\bc_2^{\text{disjoint}} & =  \roundframe{\mathfig{0.5}{definition/disjoint-blobs}} & u_i \in \ker{\text{ev}_{B_i}}

\end{align*}

\vspace{-4mm}

$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$

\end{block}

\begin{block}{}

\vspace{-5mm}

\begin{align*}

\bc_2^{\text{nested}} & = \roundframe{\mathfig{0.5}{definition/nested-blobs}} & u \in \ker{\text{ev}_{B_1}}

\end{align*}

\vspace{-4mm}

$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$

\end{block}

\end{frame}

\end{document}

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