From 33fd54309a6b4da7aebfea0a60c0d187d8e753f6 Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Fri, 18 Oct 2024 10:08:37 -0500 Subject: [PATCH] update --- talks/2024-grads/.gitignore | 4 +- talks/2024-grads/images/composition.png | Bin 0 -> 113715 bytes talks/2024-grads/{ => images}/decodecube.png | Bin talks/2024-grads/images/encode.png | Bin 0 -> 168109 bytes talks/2024-grads/images/encoding.png | Bin 0 -> 129739 bytes talks/2024-grads/images/loopspace.jpg.jpeg | Bin 0 -> 150315 bytes talks/2024-grads/images/winding.png | Bin 0 -> 147440 bytes talks/2024-grads/main.tex | 121 ++++++++++++++----- 8 files changed, 91 insertions(+), 34 deletions(-) create mode 100644 talks/2024-grads/images/composition.png rename talks/2024-grads/{ => images}/decodecube.png (100%) create mode 100644 talks/2024-grads/images/encode.png create mode 100644 talks/2024-grads/images/encoding.png create mode 100644 talks/2024-grads/images/loopspace.jpg.jpeg create mode 100644 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zrp9vN=Mw4*9m`6lVN>S-0dT@qT{otd_447c&B#BnWP1(R;p2-qnxs z)^m=5(hP`%EjglQX<*dJQ=MtIW45*m_Ww}{R#o^=-HaAo^hJ81whrPwfI&7TNA&vr zEiYB5+8p14XOYkpAVen&N55Vo5N8l68AuQ`sF1tvBs3e$^EWKz8Bf4Y%9$f|YX-nc zIpaM1m1WFK*r6#cZN-)-qH#RiXO3!8t5C=8@wQz)nu-`PQ&SQ=z5@dHf%QkC)Qvg+ z)K0{Kuz%HO1Gboe#SE_-w2N=Tx+#oxoNa?l*S)})zde_K^fNbCyj>WQ5h z?>tBVJrPJnKWnT1d|>o_#D6Zy_qF;iC0iak$4-6MkMH{N-HQG{*u=HX-qIL27Z(>K z`il*XLvM)YhsmNMxws+FLdf2b0PC7>zjJ;Vx=^P_PB*rKK>zY_!ETL-PM2l)*08My zQA*sqzFb$nCb081&c`{5LpheIor#>ce;bd79#|@>jy?2mqYEQr&~_Fl{_{ + \item Idea: define a custom type family that takes different number of loops to different "rotations" of the integers! + } + \only<3>{ + \item Define the encoding: + \begin{itemize} + \item $\mathsf{code} : S^1 \rightarrow \mathsf{Type}$ + \item $\mathsf{code}(\mathsf{base}) = \mathbb{Z}$ + \item $\mathsf{code}(\mathsf{loop}) = \mathsf{ua}(\mathsf{suc})$ + \end{itemize} + } + \end{itemize} + \end{column} + \begin{column}{.55\textwidth} + \begin{figure} + \centering + \includegraphics[width=\textwidth]{images/encoding.png} + \end{figure} + \end{column} + \end{columns} \end{frame} \begin{frame} @@ -511,10 +548,12 @@ \begin{itemize} \item Need to define the following data to prove the equivalence: + $$ (\mathsf{base} \equiv_{S^1} c) \simeq \mathsf{code}(c) $$ + \pause + \item For some $(c : S^1)$ and $(n : \mathbb{N})$ that encodes how many loopings a path to $c$ could take \begin{itemize} - \item For some $c : S^1$ and $n : \mathbb{N}$ that encodes how many loopings a path to $c$ could take is: - \item $ f : \mathsf{base} \equiv_{S^1} c \rightarrow \mathsf{code}(c) $ - \item $ g : \mathsf{base} \equiv_{S^1} c \leftarrow \mathsf{code}(c) $ + \item $ f : (\mathsf{base} \equiv_{S^1} c) \rightarrow \mathsf{code}(c) $ + \item $ g : \mathsf{code}(c) \rightarrow (\mathsf{base} \equiv_{S^1} c) $ \item $ (g \circ f)(p) \equiv \mathsf{id}_{\mathsf{base} \equiv_{S^1} c} $ \item $ (f \circ g)(n) \equiv \mathsf{id}_{\mathbb{Z}} $ \end{itemize} @@ -525,10 +564,15 @@ \frametitle{Homotopy type theory proof idea} \begin{itemize} - \item For $f : \mathsf{base} \equiv_{S^1} c \rightarrow \mathbb{Z}$, just use the winding map! - \item This allows us to compute using any loop + \item For $f : (\mathsf{base} \equiv_{S^1} c) \rightarrow \mathsf{code}(c)$, just use the winding map! + \item This allows us to turn any loop into an integer \item For some $p : \mathsf{base} \equiv \mathsf{base}$, $\mathsf{transport}^{\mathsf{code}}(p, 0)$ turns 0 into any integer by mapping over the equivalence between integers \end{itemize} + + \begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{images/encode.png} + \end{figure} \end{frame} \begin{frame} @@ -536,17 +580,19 @@ \begin{itemize} \item For $g : \mathsf{code}(c) \rightarrow \mathsf{base} \equiv_{S^1} c$, - we can iteratively compose $\mathsf{loop}$s to $\mathsf{refl}$ using a function $loop^n$ + we can iteratively compose $\mathsf{loop}$s to $\mathsf{refl}$ using a function $\mathsf{loop}^n$ \begin{itemize} - \item $loop^{-1 + n} = loop^n \cdot \mathsf{loop}^{-1}$ - \item $loop^0 = \mathsf{refl}$ - \item $loop^{n + 1} = loop^n \cdot \mathsf{loop}$ + \item $\mathsf{loop}^{-1 + n} = \mathsf{loop}^n \cdot \mathsf{loop}^{-1}$ + \item $\mathsf{loop}^0 = \mathsf{refl}$ + \item $\mathsf{loop}^{n + 1} = \mathsf{loop}^n \cdot \mathsf{loop}$ \end{itemize} + \pause \item Then, use this to define $g$ \begin{itemize} - \item $g(c : \mathsf{code}(\mathsf{base})) = loop^c$ - \item $g(c : \mathsf{code}(\mathsf{loop})) : loop^c \equiv loop^c$ + \item $g(c : \mathsf{code}(\mathsf{base})) = \mathsf{loop}^c$ + \item $g(c : \mathsf{code}(\mathsf{loop})) : (\mathsf{loop}^c \equiv^{\lambda x \mapsto \mathsf{code}(x) \rightarrow \mathsf{base} \equiv x}_{\mathsf{loop}} \mathsf{loop}^c)$ + \item This can be defined in two ways \end{itemize} \end{itemize} \end{frame} @@ -555,7 +601,7 @@ \frametitle{Homotopy type theory proof} \begin{itemize} - \item The homotopies $(g \circ f \equiv \mathsf{id}_{(\mathsf{base} \equiv_{S^1} c)})$ and $(f \circ g \equiv \mathsf{id}_{\mathsf{code}(c)})$ can be proven just by applying groupoid laws of paths: + \item The homotopies $(g \circ f \equiv \mathsf{id}_{(\mathsf{base} \equiv_{S^1} c)})$ and $(f \circ g \equiv \mathsf{id}_{\mathsf{code}(c)})$ can be proven just by applying path induction and these groupoid laws of paths: \begin{itemize} \item Identity: $(p : x \equiv_A y) \rightarrow p \cdot \mathsf{refl} \equiv p$ \item Identity: $(p : x \equiv_A y) \rightarrow \mathsf{refl} \cdot p \equiv p$ @@ -566,12 +612,21 @@ \end{itemize} \end{frame} +\begin{frame} + \frametitle{Composition} + + \begin{figure} + \centering + \includegraphics[width=0.75\textwidth]{images/composition.png} + \end{figure} +\end{frame} + \begin{frame} \frametitle{Defining $g$ with cubes} \begin{figure} \centering - \includegraphics[width=0.75\textwidth]{./decodecube.png} + \includegraphics[width=0.75\textwidth]{images/decodecube.png} \end{figure} \end{frame} @@ -589,7 +644,7 @@ \frametitle{Current work} \begin{itemize} - \item Re-formalizing results from HoTT book chapter 8 + \item Re-formalizing results from HoTT book chapters 7 on $n$-types and 8 on homotopy theory \item Re-formalizing results from Floris van Doorn's dissertation on spectral sequences \end{itemize} \end{frame} @@ -603,7 +658,7 @@ \bigskip - % \printbibliography + \printbibliography \end{frame} \end{document} \ No newline at end of file