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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}

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\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]
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\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);
}

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\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}
$$\text{ev}\Bigg(\roundframe{d \mathfig{0.12}{definition/evaluation1}} - \roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$
\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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