Début du diapo de soutenance
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Report/M2Diapo.tex
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Report/M2Diapo.tex
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% !TeX spellcheck = en_US
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\documentclass[12pt,xcolor={dvipsnames},aspectratio=169]{beamer}
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\input{./headerDiapo.tex}
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\title[Semantics of 2-sortification]{Categorical semantics of the reduction of GATs to two-sorted GATs.
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\\[1ex] \large Notes on my 4.5-month internship at the LIX}
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\hypersetup{pdftitle={Categorical semantics of the reduction of GATs to two-sorted GATs}}
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\author[Samy Avrillon]{Samy Avrillon, supervised by
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\\[1ex] Ambroise Lafont (LIX, Palaiseau, France)}
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\date{}
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\begin{document}
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\begin{frame}[plain]
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\maketitle
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\end{frame}
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\section*{Table of contents}
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\begin{frame}{Plan of the presentation}
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\tableofcontents[hidesubsections]
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\end{frame}
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\section{Introduction}
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\subsection{What is a GAT ?}
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\begin{frame}{What is a GAT ?}
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\begin{itemize}
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\item A syntactic object
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\item Defines a set of models
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\item Defines a category of models
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\end{itemize}
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\end{frame}
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\begin{frame}{A GAT for Type Theory}
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\renewcommand\to{{\;\rightarrow\;}}
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\begin{center}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{|c|c|c|}
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\hline
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$\Con : \Set$ &
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$\Ty : (\Gamma : \Con) \to \Set$ &
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$\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set$ \\\hline
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\end{tabular}
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\end{center}
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\pause
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\begin{columns}
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\begin{column}{0.60\textwidth}
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\begin{center}
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\begin{tcolorbox}[boxsep=-5pt]
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\renewcommand\arraystretch{1.25}
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\begin{tabular}{l}
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$\triangleright: (\Gamma:\Con)\to(A:\Ty\;\Gamma)\to\Con$ \\
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$\diamond: \Con$ \\
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$\operatorname{var} : (\Gamma:\Con) \to (A:\Ty\;\Gamma) \to \Tm\;(\Gamma\triangleright A)\;A$ \\
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$\implies : (\Gamma:\Con) \to \Ty\;\Gamma \to \Ty\;\Gamma \to \Ty\;\Gamma$ \\
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$\operatorname{app} : (\Gamma : \Con) \to (A\;B :\Ty\;\Gamma) \to$\\
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\qquad$\Tm\;\Gamma\;(A\implies B)\to\Tm\;\Gamma\; A \to \Tm\;\Gamma\; B$
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\end{tabular}
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\end{tcolorbox}
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\end{center}
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\end{column}
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\begin{column}{0.4\textwidth}
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\[\begin{array}{c}
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\Gamma : \Con\\
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A,B : \Ty\;\Gamma\\
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t : \Tm (\Gamma \triangleright (A \implies B)) A\\
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{\mathbf{\top}}\\
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\operatorname{app}\;\operatorname{var}\;t : \Tm\;(\Gamma\triangleright(A\implies B))\;A
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\end{array}\]
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{2-sortification}
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\renewcommand\arraystretch{1.5}
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\begin{center}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{|c|c|}
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\hline
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$\mathcal{O} : \Set$ &
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$\El : \mathcal{O} \to \Set$ \\\hline
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\end{tabular}
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\end{center}
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\todo{Cette partie}
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\end{frame}
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\begin{frame}{Category of Models \& Generalization}
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\begin{tabular}{lcp{0.5\textwidth}}
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$\boxed{\bullet}$ & $\CC_0 :=$ &
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$\one$ \\
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$\boxed{\Con : \Set}$ & $\CC_1 := $&
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\only<1>{$\left[X_\Con\right]$}
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\only<2>{$\left[X_\Con : \Set\right]$}
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\only<3>{$(\bullet : \CC_0) \times \left(\Set\right)$}
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\only<4>{$\left(X : \CC_0\right) \times \Set^{H_1(X)}$} \\
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$\boxed{\Ty : (\Gamma : \Con) \to \Set}$ & $\CC_2 := $&
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\only<1>{$\left[X_\Con, \left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con}\right]$}
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\only<2>{$\left[X_\Con : \Set, X_\Ty : \Set^{X_\Con}\right]$}
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\only<3>{$\left(X_\Con : \CC_1\right) \times \left(\Set^{X_\Con}\right)$}
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\only<4>{$\left(X : \CC_1\right) \times \Set^{H_2(X)}$} \\
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$\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}$ & $\CC_3 :=$&
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\only<1>{$\left[X_\Con, \left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con},\right.$}
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\only<2>{$\left[X_\Con : \Set, X_\Ty : \Set^{X_\Con},\right.$}
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\only<3>{$\left((X_\Con,X_\Ty) : \CC_2\right) \times$}
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\only<4>{$\left(X : \CC_2\right) \times \Set^{H_3(X)}$} \\ &&
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\only<1>{$\left.\left(\left(X_\Tm(\Delta,A)\right)_{A \in \Ty(\Delta)}\right)_{\Delta \in X_\Con} \right]$}
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\only<2>{\quad$\left.X_\Tm : \Set^{\prod_{\Delta:X_\Con}X_\Ty(\Delta)}\right]$}
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\only<3>{\quad$\left(\Set^{\prod_{\Delta:X_\Con}X_\Ty(\Delta)}\right)$}
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\end{tabular}
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\pause[3]
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\[\begin{array}{rcl}
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H_1(\bullet) &=& 1_\Set \\
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H_2(X_\Con) &=& X_\Con \\
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H_3(X_\Con,X_\Ty) &=& \displaystyle\prod_{\Delta:X_\Con}X_\Ty(\Delta)
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\end{array}\]
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\end{frame}
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\begin{frame}{Category of Models of transformed GAT}
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\[
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\begin{array}{|c|}
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\hline
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\mathcal{O} : \Set \\
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\El : \mathcal{O} \to \Set \\
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\hline
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\end{array}
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\]
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\[
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\BB_0 := \left(X_\UU : \Set, X_\El : \Set^{X_\UU}\right)
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\uncover<2->{\ensuremath{\simeq \left(X_\UU : \Set, X_\El : \Set/X_\UU\right)}}
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\uncover<3->{\ensuremath{\equiv \TSet}}
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\]
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\pause
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\[
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\TT := \simpleUpDownArrow{El}{p}{\UU}
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\]
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\pause
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\begin{center}
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\begin{tabular}{ll}
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$X_\UU$:& Sortes \\
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$X_\El$:& Objets\\
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$X_p(x)$:& Sorte de l'objet \\
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\end{tabular}
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\end{center}
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\end{frame}
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\begin{frame}{Adding transformed sort declarations}
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\begin{center}
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\begin{tabular}{ll}
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$X_\UU$:& Sortes \\
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$X_\El$:& Objets\\
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$X_p(x)$:& Sorte de l'objet \\
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\end{tabular}
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\end{center}
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\begin{center}
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\begin{tabular}{lrp{0.4\textwidth}}
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$\boxed{\mathcal{O} : \Set\quad\El : \mathcal{O} \to \Set}$ & $\BB_0 =$ &
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$X : \TSet$ \\
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$\boxed{\Con : \mathcal{O}}$ & $\Cstr_\Con :$ &
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\only<1>{$X_\UU$}
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\only<2>{$H_1F_0(X) \to X_\UU$} \\
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$\boxed{\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}}$ & $\Cstr_\Ty :$&
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\only<1>{$\left(\Gamma \in X_\El\middle|X_p(\Gamma) = \Cstr_\Con\right) \to X_\UU $}
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\only<2>{$H_2F_1(X,\Cstr_\Con) \to X_\UU$} \\
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$\boxed{\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}}$ & $\Cstr_\Tm : $ &
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\only<1>{$\begin{array}{c}\left(\Delta \in X_\El\middle|X_p(\Delta) = \Cstr_\Con\right) \to\\ \left(A \in X_\El\middle|X_p(A) = \Cstr_\Ty(\Delta)\right) \to\\ X_\UU\end{array}$}
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\only<2>{$H_3F_2(X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU$}
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\end{tabular}
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\end{center}
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\end{frame}
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\begin{frame}{Constructing the functors}
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\[
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\begin{array}{l|l}
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Y : \mathbf{\CC_3} & X : \mathbf{\BB_3} \\\hline
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& X : \TSet\\
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Y_\Con : \Set & \Cstr_\Con : X_\UU \\
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\left(Y_\Ty(\Gamma)\right)_{\Gamma \in Y_\Con} &
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\Cstr_\Ty : H_2 F_{1} (X,\Cstr_\Con) \to X_\UU \\
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\left(\left(Y_\Tm(\Delta,A)\right)_{A \in Y_\Ty(\Delta)}\right)_{\Delta \in Y_\Con} &
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\Cstr_\Tm : H_3 F_{2} (X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU \\
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\end{array}
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\]
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\begin{center}
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\begin{tabular}{ll}
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$X_\UU$:& Sortes \\
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$X_\El$:& Objets\\
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$X_p(x)$:& Sorte de l'objet \\
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\end{tabular}
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\end{center}
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\end{frame}
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\begin{frame}{Constructing $G_3$}
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\only<1>{
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\[\begin{array}{ccl}
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X_\El & = & \text{«objets»}\\
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\labeledupdownarrow{p}&& \labeledupdownarrow{\text{«sorte de l'objet»}}\\
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X_\UU & = & \text{«sortes»}
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\end{array}\]
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}
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\pause
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\[\begin{array}{ccccccc}
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X_\El & = &
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Y_\Con & \oplus &
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\displaystyle\coprod_{\Gamma \in Y_\Con}Y_\Ty(\Gamma) & \oplus &
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\displaystyle\coprod_{\Delta \in Y_\Con}\coprod_{A \in Y_\Ty(\Delta)}Y_\Ty(\Delta,A)\\
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\labeledupdownarrow{p}&&\labeledupdownarrow{}&&\labeledupdownarrow{\operatorname{proj}_1}&&\labeledupdownarrow{\operatorname{proj}_{2,1}}\\
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X_\UU & = &
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1 & \oplus &
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Y_\Con & \oplus&
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\displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta)
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\end{array}\]
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\pause
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\[\begin{array}{lcl}
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\Cstr^X_\Con &=& \inj_1(\star)\\
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\Cstr^X_\Ty(\Gamma) &=& \inj_2(\Gamma) \\
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\Cstr^X_\Tm(\Delta,A) &=& \inj_3(\Delta,A)
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\end{array}\]
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\end{frame}
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\begin{frame}{Constructing $F_3$}
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\begin{center}
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\begin{tabular}{ll}
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$X_\UU$:& Sortes \\
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$X_\El$:& Objets\\
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$X_p(x)$:& Sorte de l'objet \\
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\end{tabular}
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\end{center}
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\renewcommand\arraystretch{1.8}
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\[
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\begin{array}{l|l}
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X : \mathbf{\BB_3} & Y = F_3(X): \mathbf{\CC_3}\\\hline
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X : \TSet &\\
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\Cstr_\Con : X_\UU & Y_\Con =
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\uncover<2->{\ensuremath{X_p^{-1}(\left\{\Cstr^X_\Con\right\}) \subseteq X_\El}}\\
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\Cstr_\Ty : H_2 F_{1} (X,\Cstr_\Con) \to X_\UU &
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Y_\Ty(\Gamma) =
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\uncover<3->{\ensuremath{X_p^{-1}(\{\Cstr^X_\Ty(\Gamma)\})}}\\
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\Cstr_\Tm : H_3 F_{2} (X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU &
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Y_\Tm(\Delta,A) =
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\uncover<4->{\ensuremath{X_p^{-1}(\Cstr^X_\Tm(\Delta,A))}} \\
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\end{array}
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\]
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\end{frame}
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\end{document}
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% !TeX spellcheck = en_US
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% !TeX spellcheck = en_US
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\documentclass[11pt,a4paper]{article}
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\documentclass[11pt,a4paper]{article}
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\usepackage[textheight=0.75\paperheight,a4paper,margin=2cm]{geometry}
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\input{./header.tex}
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\input{./header.tex}
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\usepackage{titling}
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% po4a: environment remark
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% po4a: environment remark
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% po4a: environment tikzpicture
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% po4a: environment property
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% po4a: environment property
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\usepackage[backend=biber,style=numeric]{biblatex}
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\usepackage[backend=biber,style=numeric]{biblatex}
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\usepackage{hyperref}
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\usepackage[textheight=0.75\paperheight,a4paper,margin=2cm]{geometry}
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\usepackage[lighttt]{lmodern}
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% Création des environnement globaux
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% Création des environnement globaux
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\newtheorem{theorem}{Theorem}
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\newtheorem{definition}{Definition}
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\newcommand\todo[1]{\begin{tcolorbox}[colback=red!20!white,colframe=red!85!black,boxrule=4pt] #1 \end{tcolorbox}}
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\newcommand\todo[1]{\begin{tcolorbox}[colback=red!20!white,colframe=red!85!black,boxrule=4pt] #1 \end{tcolorbox}}
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\newcommand\inlinetodo[1]{\colorbox{orange}{#1}}
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\newcommand\inlinetodo[1]{\colorbox{orange}{#1}}
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% Création des environnements spécifiques au document
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% Création des environnements spécifiques au document
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\fancyfoot[R]{\textbf{\thepage / \pageref{LastReportPage}}}
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\fancyfoot[R]{\textbf{\thepage / \pageref{LastReportPage}}}
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\usepackage{titling}
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\addbibresource{Bilibibio.bib}
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\addbibresource{Bilibibio.bib}
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% Loading packages
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\usepackage{fontspec}
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\usepackage{cprotect}
|
||||||
|
\usepackage{xpatch}
|
||||||
|
\usepackage{enumitem}
|
||||||
|
\usepackage{amsmath}
|
||||||
|
\usepackage{amsfonts}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{txfonts}
|
||||||
|
\usepackage{yade}
|
||||||
|
\usepackage{environ}
|
||||||
|
\usepackage[backend=biber,style=numeric]{biblatex}
|
||||||
|
\usepackage{hyperref}
|
||||||
|
\usepackage{array}
|
||||||
|
|
||||||
|
\usepackage[lighttt]{lmodern}
|
||||||
|
\usetikzlibrary{shapes.geometric,positioning,backgrounds}
|
||||||
|
|
||||||
|
% Macros caractères globales
|
||||||
|
\newcommand{\pgrph}{\P}
|
||||||
|
\newcommand{\hsep}{\vspace{.2cm}\centerline{\rule{0.8\linewidth}{.05pt}}\vspace{.4cm}}
|
||||||
|
\renewcommand{\P}{\mathbb{P}}
|
||||||
|
\newcommand{\E}{\mathbb{E}}
|
||||||
|
\newcommand{\1}{\scalebox{1.2}{$\mathbbm{1}$}}
|
||||||
|
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
|
||||||
|
\newcommand{\littleO}{o}
|
||||||
|
\newcommand{\bigO}{\mathcal{O}}
|
||||||
|
\newcommand{\longdash}{\:\textrm{---}\:}
|
||||||
|
\newcommand\hole{\left[\raisebox{-0.25ex}{\scalebox{1.2}{$\cdot$}}\right]}
|
||||||
|
\newcommand\bracket[1]{\!\left[#1\right]}
|
||||||
|
\newcommand\spacebar{\;|\;}
|
||||||
|
\def\nDownarrow{\not\mspace{1mu}\Downarrow}
|
||||||
|
\let\pprec\preccurlyeq
|
||||||
|
|
||||||
|
% Création des environnement globaux
|
||||||
|
\newtheorem{property}{Property}
|
||||||
|
\newtheorem{remark}{Remark}
|
||||||
|
|
||||||
|
\newcounter{rule}
|
||||||
|
|
||||||
|
% Commandes logiques globales
|
||||||
|
\newcommand{\ifnullthenelse}[3]{
|
||||||
|
\ifnum\value{#1}=0
|
||||||
|
#2
|
||||||
|
\else
|
||||||
|
#3
|
||||||
|
\fi
|
||||||
|
}
|
||||||
|
|
||||||
|
%%% Commande \newtag permettant de changer le label d'une equation
|
||||||
|
\makeatletter
|
||||||
|
\newcommand\newtag[2]{#1\def\@currentlabel{#1}\label{#2}}
|
||||||
|
\makeatother
|
||||||
|
|
||||||
|
|
||||||
|
% Macros caractères spécifiques au document
|
||||||
|
\newfontface\russian{Liberation Serif}
|
||||||
|
\newcommand\BB{{\ensuremath{\mathcal{B}}}}
|
||||||
|
\newcommand\en{{\text{\russian н}}}
|
||||||
|
\newcommand\TT{{\ensuremath{\mathcal{T}}}}
|
||||||
|
\newcommand\UU{{\ensuremath{\mathcal{U}}}}
|
||||||
|
\newcommand\CC{{\ensuremath{\mathcal{C}}}}
|
||||||
|
\newcommand\El{{\ensuremath{\operatorname{\mathcal{E}l}}}}
|
||||||
|
\newcommand\ii{{\ensuremath{\mathbf{i}}}}
|
||||||
|
\newcommand\Con{{\ensuremath{\operatorname{Con}}}}
|
||||||
|
\newcommand\Ty{{\ensuremath{\operatorname{Ty}}}}
|
||||||
|
\newcommand\Tm{{\ensuremath{\operatorname{Tm}}}}
|
||||||
|
\newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}
|
||||||
|
\newcommand\Rtsc{{\ensuremath{\operatorname{\mathcal{R}tsc}}}}
|
||||||
|
\newcommand\Cat{{\ensuremath{\operatorname{\mathcal{C}at}}}}
|
||||||
|
\newcommand\Set{{\ensuremath{\operatorname{\mathcal{S}et}}}}
|
||||||
|
\newcommand\FamSet{{\ensuremath{\operatorname{\mathcal{F}am\mathcal{S}et}}}}
|
||||||
|
\newcommand\Hom{{\ensuremath{\operatorname{\mathcal{H}om}}}}
|
||||||
|
\newcommand\this{{\ensuremath{\operatorname{\texttt{this}}}}}
|
||||||
|
\newcommand\one{{\ensuremath{\mathbf{1}}}}
|
||||||
|
\newcommand\dash{{\;\textrm{---}\;}}
|
||||||
|
\renewcommand\enquote[1]{``#1''}
|
||||||
|
\newcommand\tl{{\triangleleft}}
|
||||||
|
|
||||||
|
\DeclareMathOperator{\inj}{inj}
|
||||||
|
\DeclareMathOperator{\id}{id}
|
||||||
|
\DeclareMathOperator{\Id}{\mathcal{I}d}
|
||||||
|
|
||||||
|
\newcommand\TSet{{\ensuremath{\left[\TT,\Set\right]}}}
|
||||||
|
\newcommand\TSetObject[3]{{\ensuremath{
|
||||||
|
\left[
|
||||||
|
\begin{tikzpicture}[baseline=(base)]
|
||||||
|
\node (U) at (0,0) {\UU};
|
||||||
|
\node (El) at (0,1) {\El};
|
||||||
|
\draw[->] (El) -- node[anchor=east] {\ensuremath{\scriptstyle p}} (U);
|
||||||
|
|
||||||
|
\node (XU) at (1.5,0) {\ensuremath{#3}};
|
||||||
|
\node (XEl) at (1.5,1) {\ensuremath{#1}};
|
||||||
|
\draw[->] (XEl) -- node[anchor=west] {\ensuremath{\scriptstyle #2}} (XU);
|
||||||
|
|
||||||
|
\draw[|->] (.3,0) -- (1,0);
|
||||||
|
\draw[|->] (.3,0.5) -- (1,0.5);
|
||||||
|
\draw[|->] (.3,1) -- (1,1);
|
||||||
|
|
||||||
|
\coordinate (base) at (.5,.4);
|
||||||
|
\end{tikzpicture}
|
||||||
|
\right]
|
||||||
|
}}}
|
||||||
|
|
||||||
|
\newcommand\simpleArrow[3]{
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node (0) at (0,0) {$#1$};
|
||||||
|
\node (1) at (6,0) {$#3$};
|
||||||
|
\path[->] (0) edge["${\scriptstyle #2}$", pos=0.5, fore, black,=>, ] (1);
|
||||||
|
\end{tikzpicture}
|
||||||
|
}
|
||||||
|
\newcommand\simpleUpDownArrow[3]{
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node (0) at (0,1) {$#1$};
|
||||||
|
\node (1) at (0,0) {$#3$};
|
||||||
|
\path[->] (0) edge["${\scriptstyle #2}$", pos=0.5, fore, black,=>, ] (1);
|
||||||
|
\end{tikzpicture}
|
||||||
|
}
|
||||||
|
\newcommand\labeledupdownarrow[1]{
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\path[->] (0,1) edge["${\scriptstyle #1}$", pos=0.5, fore, black,=>, ] (0,0);
|
||||||
|
\end{tikzpicture}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newcommand\diagram[1]{\begin{tcolorbox}\begin{center}\vspace{1.5cm}\Huge \texttt{\textbf{#1}}\vspace{1.5cm}\end{center}\end{tcolorbox}}
|
||||||
|
|
||||||
|
\newcommand\todo[1]{\begin{tcolorbox}[colback=red!20!white,colframe=red!85!black,boxrule=4pt] #1 \end{tcolorbox}}
|
||||||
|
\newcommand\inlinetodo[1]{\colorbox{orange}{#1}}
|
||||||
|
|
||||||
|
\newcommand{\backupbegin}{
|
||||||
|
\newcounter{finalframe}
|
||||||
|
\setcounter{finalframe}{\value{framenumber}}
|
||||||
|
}
|
||||||
|
\newcommand{\backupend}{
|
||||||
|
\setcounter{framenumber}{\value{finalframe}}
|
||||||
|
}
|
||||||
|
|
||||||
|
\addbibresource{Bilibibio.bib}
|
||||||
|
|
||||||
|
\usetheme{Madrid}
|
||||||
|
|
||||||
|
\makeatletter
|
||||||
|
\setbeamertemplate{frametitle}{%
|
||||||
|
\nointerlineskip
|
||||||
|
\begin{beamercolorbox}[sep=1.6ex,wd=\paperwidth,leftskip=.5cm,rightskip=0cm]{frametitle}%
|
||||||
|
\usebeamerfont{frametitle}\usebeamercolor[fg]{frametitle}\beamer@frametitle\\
|
||||||
|
\usebeamerfont{framesubtitle}\usebeamercolor[fg]{framesubtitle}\insertframesubtitle
|
||||||
|
\end{beamercolorbox}%
|
||||||
|
}
|
||||||
|
\makeatother
|
||||||
|
|
||||||
|
\hypersetup{pdfpagemode=FullScreen}
|
||||||
|
% Transition en fade-in par défaut
|
||||||
|
\addtobeamertemplate{background canvas}{\transfade[duration=0.4]}{}
|
||||||
|
|
||||||
|
\addtobeamertemplate{frametitle}{
|
||||||
|
\let\insertframetitle\insertsubsectionhead}{}
|
||||||
|
\makeatletter
|
||||||
|
\CheckCommand*\beamer@checkframetitle{\@ifnextchar\bgroup\beamer@inlineframetitle{}}
|
||||||
|
\renewcommand*\beamer@checkframetitle{\global\let\beamer@frametitle\relax\@ifnextchar\bgroup\beamer@inlineframetitle{}}
|
||||||
|
\makeatother
|
||||||
|
|
||||||
|
\setbeamertemplate{itemizeitem}{\scriptsize$\diamond$}
|
||||||
|
\newcommand\sectocframe[1][]{
|
||||||
|
\begin{frame}
|
||||||
|
\pdfpcnote{#1}
|
||||||
|
\tableofcontents[currentsection,hideothersubsections,sections=\value{section}]
|
||||||
|
\end{frame}
|
||||||
|
}
|
||||||
Loading…
x
Reference in New Issue
Block a user