262 lines
7.9 KiB
TeX
262 lines
7.9 KiB
TeX
% !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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