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 $\cC$.

   104 \end{block}

   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}