equal
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51 transparent=5, |
51 transparent=5, |
52 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, |
52 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, |
53 again covered={\opaqueness<1->{50}} |
53 again covered={\opaqueness<1->{50}} |
54 } |
54 } |
55 |
55 |
|
56 \uncover<2>{ |
56 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
57 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
57 \uncover<2>{ |
|
58 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; |
58 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; |
59 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
59 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
60 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
60 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
61 } |
61 } |
62 |
62 |
77 |
77 |
78 \section{TQFTs} |
78 \section{TQFTs} |
79 |
79 |
80 \begin{frame}{$n$-categories} |
80 \begin{frame}{$n$-categories} |
81 \begin{block}{There are many definitions of $n$-categories!} |
81 \begin{block}{There are many definitions of $n$-categories!} |
82 For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal $2$-categories. |
82 For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal categories. |
83 \end{block} |
83 \end{block} |
84 \begin{block}{We have another definition: \emph{topological $n$-categories}} |
84 \begin{block}{We have yet another definition: \emph{topological $n$-categories}} |
85 \begin{itemize} |
85 \begin{itemize} |
86 %\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. |
86 %\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. |
87 \item A vector space $\cC(B^n)$ for every $n$-ball $B$. |
87 \item A vector space $\cC(B^n)$ for every $n$-ball $B$. |
88 %\item From these, inductively |
88 %\item From these, inductively |
89 %\begin{itemize} |
89 %\begin{itemize} |
144 \vspace{4mm} |
144 \vspace{4mm} |
145 %\begin{itemize} |
145 %\begin{itemize} |
146 %\item We can also associate a $k$-category to an $n-k$-manifold. |
146 %\item We can also associate a $k$-category to an $n-k$-manifold. |
147 %\item We don't assign a number to an $n+1$-manifold (a `decapitated' extended TQFT). |
147 %\item We don't assign a number to an $n+1$-manifold (a `decapitated' extended TQFT). |
148 %\end{itemize} |
148 %\end{itemize} |
149 $\cA(Y \times [0,1])$ is a $1$-category, and when $Y \subset \bdy X$, $\cA(X)$ is a module over $\cA(Y \times [0,1])$. |
149 $\cA(Y^{n-1} \times [0,1])$ is a $1$-category, and when $Y \subset \bdy X$, $\cA(X)$ is a module over $\cA(Y \times [0,1])$. |
150 \begin{thm}[Gluing formula] |
150 \begin{thm}[Gluing formula] |
151 When $Y \sqcup Y^{\text{op}} \subset \bdy X$, |
151 When $Y \sqcup Y^{\text{op}} \subset \bdy X$, |
152 \vspace{-1mm} |
152 \vspace{-1mm} |
153 \[ |
153 \[ |
154 \cA(X \bigcup_Y \selfarrow) \iso \cA(X) \bigotimes_{\cA(Y \times [0,1])} \selfarrow. |
154 \cA(X \bigcup_Y \selfarrow) \iso \cA(X) \bigotimes_{\cA(Y \times [0,1])} \selfarrow. |
334 \end{frame} |
334 \end{frame} |
335 |
335 |
336 \mode<beamer>{ |
336 \mode<beamer>{ |
337 \begin{frame}{An action of $\CH{\cM}$} |
337 \begin{frame}{An action of $\CH{\cM}$} |
338 \begin{proof} |
338 \begin{proof} |
339 \begin{description} |
339 Uniqueness: |
340 \item[Step 1] If $\cM=B^n$ or a union of balls, there's a unique chain map, since $\bc_*(B^n; \cC) \htpy \cC$ is concentrated in homological degree $0$. |
340 \begin{description} |
341 \item[Step 2] Fix an open cover $\cU$ of balls. \\ A family of homeomorphisms $P^k \to \Homeo(\cM)$ can be broken up in into pieces, each of which is supported in at most $k$ open sets from $\cU$. \qedhere |
341 \item[Step 1] If $\cM=B^n$ or a union of balls, there's a unique (up to homotopy) chain map, since $\bc_*(B^n; \cC) \htpy \cC$ is concentrated in homological degree $0$. |
|
342 \item[Step 2] Fix an open cover $\cU$ of balls. \\ A family of homeomorphisms $P^k \to \Homeo(\cM)$ can be broken up in into pieces, each of which is supported in at most $k$ open sets from $\cU$. \end{description} |
|
343 Existence: |
|
344 \begin{description} |
|
345 \item[Step 3] Show that all of the choices available above can be made consistently, using the method of acyclic models. \qedhere |
342 \end{description} |
346 \end{description} |
343 \end{proof} |
347 \end{proof} |
344 \end{frame} |
348 \end{frame} |
345 } |
349 } |
346 |
350 |