% !TeX spellcheck = en_US \DocumentMetadata{} \documentclass[12pt,xcolor={dvipsnames},aspectratio=169]{beamer} \input{./headerDiapo.tex} \title[Semantics of 2-sortification]{Categorical semantics of the reduction of GATs to two-sorted GATs. \\[1ex] \large Notes on my 4.5-month internship at the LIX} \hypersetup{pdftitle={Categorical semantics of the reduction of GATs to two-sorted GATs}} \author[Samy Avrillon]{Samy Avrillon, supervised by \\[1ex] Ambroise Lafont (LIX, Palaiseau, France)} \date{} \begin{document} \begin{frame}[plain] \maketitle \end{frame} \section*{Table of contents} \begin{frame}{Plan of the presentation} \tableofcontents[hidesubsections] \end{frame} \section{GATs and 2-sortification} \begin{frame}{What is a GAT ?} \begin{itemize} \item A dependent \textbf{type theory} \item[\ding{220}] Enables to create sorts, object of those sorts, equalities between those objects \item A \textbf{syntactic object} \item[\ding{220}] Type judgements are defined with induction rules \item Describes \textbf{models} \item[\ding{220}] Defines a category of models \end{itemize} \end{frame} \begin{frame}{A GAT for a function in Set} \begin{tcolorbox} \begin{columns} \begin{column}{.6\textwidth} \renewcommand\arraystretch{1.5} \begin{tabular}{l} $A : \Set$ \\ $B : \Set$ \\\hline $\operatorname{exec} : A \to B$ \end{tabular} \end{column} \begin{column}{.4\textwidth} Models: triples (A,B,f) \end{column} \end{columns} \end{tcolorbox} \end{frame} \begin{frame}{A GAT for an bijective function in Set} \begin{tcolorbox} \begin{columns} \begin{column}{.6\textwidth} \renewcommand\arraystretch{1.5} \begin{tabular}{l} $A : \Set$ \\ $B : \Set$ \\\hline $\operatorname{exec} : A \to B$ \\ $\operatorname{invexec} : B \to A$\\\hline $\operatorname{isol} : (x : A) \to \operatorname{invexec}(\operatorname{exec}\;x) = x$\\ $\operatorname{isor} : (y : B) \to \operatorname{exec}(\operatorname{invexec}\;y) = y$\\ \end{tabular} \end{column} \begin{column}{.4\textwidth} Models: triples (A,B,f) s.t. f is bijective \end{column} \end{columns} \end{tcolorbox} \end{frame} \begin{frame}{A GAT for a small category} \begin{tcolorbox} \begin{columns} \begin{column}{0.8\textwidth} \renewcommand\arraystretch{1.5} \begin{tabular}{l} $\Obj : \Set$ \\ $\Hom : \Obj \to \Obj \to \Set$ \\\hline $\id : (A : \Obj) \to \Hom\;A\;A$ \\ $\operatorname{comp} : (A\;B\;C:\Obj) \to \Hom\;B\;C \to\Hom\;A\;B \to \Hom\;A\;C$ \\\hline $\operatorname{idl}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \operatorname{comp} (\id\;B) \sigma = \sigma$\\ $\operatorname{idr}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \operatorname{comp} \sigma (\id\;A) = \sigma$\\ $\operatorname{comp-trans}: \dots$\\ \end{tabular} \end{column} \begin{column}{0.2\textwidth} Models: small categories \end{column} \end{columns} \end{tcolorbox} \end{frame} \begin{frame}{A GAT for Type Theory} \begin{tcolorbox} \begin{columns} \begin{column}{0.7\textwidth} \renewcommand\arraystretch{1.5} \begin{tabular}{l} $\Con : \Set$ \\ $\Ty : \Con \to \Set$ \\ $\Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set$ \\\hline $\operatorname{empty}: \Con$ \\ $\operatorname{ext}: (\Gamma:\Con)\to(A:\Ty\;\Gamma)\to\Con$ \\ $\operatorname{implies} : (\Gamma:\Con) \to \Ty\;\Gamma \to \Ty\;\Gamma \to \Ty\;\Gamma$ \\ $\operatorname{app} : (\Gamma : \Con) \to (A\;B :\Ty\;\Gamma) \to$\\ \qquad$\Tm\;\Gamma\;(\operatorname{implies}\;A\;B)\to\Tm\;\Gamma\; A \to \Tm\;\Gamma\; B$ \end{tabular} \end{column} \begin{column}{0.3\textwidth} Models : Triples $(X_\Con,X_\Ty,X_\Tm)$ with constructors $\operatorname{empty} \in X_\Con$ \qquad\vdots \end{column} \end{columns} \end{tcolorbox} \end{frame} \begin{frame}{2-sortification} \begin{center} {\large Transform a GAT into a GAT with only two sorts} \ding{229} Simplify a study of GATs \end{center} \end{frame} \begin{frame}{2-sortification of the Set Function GAT} \begin{tcolorbox} \renewcommand\arraystretch{1.5} \begin{tabular}{llrcl} \pause $\mathcal{O} : \Set$ & \color{RoyalBlue} \text{sorts} & $o:\mathcal{O}$ &$\leftrightarrow$&$o$ is a sort \\ $\El : \mathcal{O} \to \Set$ & \color{RoyalBlue}\text{objects of that sort} & \qquad $x : \El\;o$ & $\leftrightarrow$ & $x : \mathcal{O}$ \pause\\\hline $A : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$A : \Set$}&&\\ $B : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$B : \Set$}&&\\ $\operatorname{exec} : \El\;A \to \El\;B$&&\color{teal}$\operatorname{exec} : A \to B$&& \end{tabular} \end{tcolorbox} \end{frame} \begin{frame}{2-sortification of Type Theory GAT} \begin{tcolorbox} \renewcommand\arraystretch{1.4} \begin{tabular}{l} $\mathcal{O} : \Set$ \\ $\El : \mathcal{O} \to \Set$ \pause\\\hline $\Con : \mathcal{O}$ \\ $\Ty : \El\;\Con \to \mathcal{O}$ \\ $\Tm : (\Gamma : \El\;\Con) \to \El\;(\Ty\;\Gamma) \to \mathcal{O}$ \\ $\operatorname{empty}: \El\;\Con$ \\ $\operatorname{ext}: (\Gamma:\El\;\Con)\to(A:\El\;(\Ty\;\Gamma))\to\El\;\Con$ \\ $\operatorname{implies} : (\Gamma:\El\;\Con) \to \El\;(\Ty\;\Gamma) \to \El\;(\Ty\;\Gamma) \to \El\;(\Ty\;\Gamma)$ \end{tabular} \end{tcolorbox} \end{frame} \begin{frame}{Goal of the internship}{} {\large Is this transformation correct ?} \ding{229} Can one study all GATs by studying only GATs with two sorts \vspace{1ex} {\large How to state this fact} \ding{229} Semantical proof \begin{center} \includesvg[scale=.4]{graphs/diagrammeFG.svg} \end{center} \ding{229} This adjunction proves that one can make the initial model of any GAT from the initial model of the transformed GAT \end{frame} \section{One Example} \begin{frame}{Constructing the categories} \begin{center} \only<1>{GAT = sorts + constructors + equalities} \only<2>{GAT = sorts + \sout{constructors} + \sout{equalities}} \end{center} \pause[2] \begin{center} \begin{tabular}{|l|l|} \hline $\begin{array}{l} \Con : \Set \\ \Ty : \Con \to \Set \\ \Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set \end{array}$ & $\begin{array}{l} \mathcal{O} : \Set\\ \El : \mathcal{O} \to \Set \\\hline \underline{\Con} : \mathcal{O} \\ \underline{\Ty} : \El\;\underline{\Con} \to \mathcal{O} \\ \underline{\Tm} : (\Gamma : \El\;\underline{\Con}) \to \El(\underline{\Ty}\;\Gamma) \to \mathcal{O} \end{array}$\\\hline\rule{0pt}{1.0\normalbaselineskip} \only<2>{$\CC \hookrightarrow (\Con,\Ty,\Tm)$} \only<3->{$\CC_0 \hookrightarrow ()$} & \only<2>{$\BB \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty},\underline{\Tm})$} \only<3->{$\BB_0 \hookrightarrow (\mathcal{O},\El)$}\\ \uncover<3->{$\CC_1 \hookrightarrow (\Con)$} & \uncover<3->{$\BB_1 \hookrightarrow (\mathcal{O},\El,\underline{\Con})$}\\ \uncover<3->{$\CC_2 \hookrightarrow (\Con,\Ty)$} & \uncover<3->{$\BB_2 \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty})$}\\ \uncover<3->{$\CC_3 \hookrightarrow (\Con,\Ty,\Tm)$} & \uncover<3->{$\BB_3 \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty},\underline{\Tm})$}\\\hline \end{tabular} \end{center} \end{frame} \begin{frame}{Category of Models \& Generalization} \begin{tabular}{lcp{0.5\textwidth}} $\boxed{()}$ & $\CC_0 :=$ & $\one$ \\ $\boxed{\Con : \Set}$ & $\CC_1 := $& \only<1>{$\left[X_\Con\right]$} \only<2>{$\left[X_\Con : \Set\right]$} \only<3>{$(\bullet : \CC_0) \times \left(\Set\right)$} \only<4>{$\left(X : \CC_0\right) \times \Set^{H_1(X)}$} \\ $\boxed{\Ty : (\Gamma : \Con) \to \Set}$ & $\CC_2 := $& \only<1>{$\left[X_\Con, \left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con}\right]$} \only<2>{$\left[X_\Con : \Set, X_\Ty : \Set^{X_\Con}\right]$} \only<3>{$\left(X_\Con : \CC_1\right) \times \left(\Set^{X_\Con}\right)$} \only<4>{$\left(X : \CC_1\right) \times \Set^{H_2(X)}$} \\ $\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}$ & $\CC_3 :=$& \only<1>{$\left[X_\Con, \left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con},\right.$} \only<2>{$\left[X_\Con : \Set, X_\Ty : \Set^{X_\Con},\right.$} \only<3>{$\left((X_\Con,X_\Ty) : \CC_2\right) \times$} \only<4>{$\left(X : \CC_2\right) \times \Set^{H_3(X)}$} \\ && \only<1>{$\left.\left(\left(X_\Tm(\Delta,A)\right)_{A \in \Ty(\Delta)}\right)_{\Delta \in X_\Con} \right]$} \only<2>{\quad$\left.X_\Tm : \Set^{\prod_{\Delta:X_\Con}X_\Ty(\Delta)}\right]$} \only<3>{\quad$\left(\Set^{\prod_{\Delta:X_\Con}X_\Ty(\Delta)}\right)$} \end{tabular} \pause[4] \[\begin{array}{rcl} H_1(\bullet) &=& 1_\Set \\ H_2(X_\Con) &=& X_\Con \\ H_3(X_\Con,X_\Ty) &=& \displaystyle\prod_{\Delta:X_\Con}X_\Ty(\Delta) \end{array}\] \end{frame} \begin{frame}{Category of Models of transformed GAT} \[ \begin{array}{|c|} \hline \mathcal{O} : \Set \\ \El : \mathcal{O} \to \Set \\ \hline \end{array} \] \[ \BB_0 := \left(X_\UU : \Set, X_\El : \Set^{X_\UU}\right) \] \pause \begin{center} \begin{tabular}{ll} $X_\UU$:& Sorts \\ $X_\El(o)$:& Objects of sort $o$\\ \end{tabular} \end{center} \end{frame} \begin{frame}{Adding transformed sort declarations} \begin{center} \begin{tabular}{ll} $X_\UU$:& Sorts \\ $X_\El(o)$:& Objects of sort $o$\\ \end{tabular} \end{center} \begin{center} \renewcommand\arraystretch{1.4} \begin{tabular}{lrp{0.4\textwidth}} $\boxed{\mathcal{O} : \Set\quad\El : \mathcal{O} \to \Set}$ & $\BB_0 =$ & $(X_\UU,X_\El)$ \\ $\boxed{\Con : \mathcal{O}}$ & $\Cstr_\Con :$ & \only<1>{$X_\UU$} \only<2>{$H_1F_0(X) \to X_\UU$} \\ $\boxed{\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}}$ & $\Cstr_\Ty :$& \only<1>{$\left(\Gamma \in X_\El(\Cstr_\Con)\right) \to X_\UU $} \only<2>{$H_2F_1(X,\Cstr_\Con) \to X_\UU$} \\ $\boxed{\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}}$ & $\Cstr_\Tm : $ & \only<1>{\rule[6ex]{0pt}{0pt}\renewcommand\arraystretch{0.9}$\begin{array}{c}\left(\Delta \in X_\El(\Cstr_\Con)\right) \to\\ \left(A \in X_\El(\Cstr_\Ty(\Delta))\right) \to\\ X_\UU\end{array}$} \only<2>{$H_3F_2(X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU$} \end{tabular} \end{center} \end{frame} \begin{frame}{Constructing the functors} \[ \begin{array}{l|l} Y : \mathbf{\CC_3} & X : \mathbf{\BB_3} \\\hline & X_\UU : \Set \quad X_\El : \Set^{X_\UU}\\ Y_\Con : \Set & \Cstr_\Con : X_\UU \\ \left(Y_\Ty(\Gamma)\right)_{\Gamma \in Y_\Con} & \Cstr_\Ty : H_2 F_{1} (X,\Cstr_\Con) \to X_\UU \\ \left(\left(Y_\Tm(\Delta,A)\right)_{A \in Y_\Ty(\Delta)}\right)_{\Delta \in Y_\Con} & \Cstr_\Tm : H_3 F_{2} (X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU \\ \end{array} \] \begin{center} \begin{tabular}{ll} $X_\UU$:& Sorts \\ $X_\El(o)$:& Objects of sort $o$\\ \end{tabular} \end{center} \end{frame} \begin{frame}{Constructing $G_3$} \only<1>{ \[\begin{array}{ccl} X_\UU & = & \text{«sorts»}\\ X_\El(o) & = & \text{«objects of sort $o$»} \end{array}\] } \pause \[\begin{array}{ccccccc} X_\UU & = & 1 & \oplus & Y_\Con & \oplus& \displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta) \\ X_\El(\inj_1(\star)) & = & Y_\Con &&&&\\ X_\El(\inj_2(\Gamma)) & = & &&Y_\Ty(\Gamma) &&\\ X_\El(\inj_3(\Delta,A)) & = & &&&& Y_\Tm(\Delta,A)\\ \end{array}\] \pause \[\begin{array}{lcl} \Cstr_\Con &=& \inj_1(\star)\\ \Cstr_\Ty(\Gamma) &=& \inj_2(\Gamma) \\ \Cstr_\Tm(\Delta,A) &=& \inj_3(\Delta,A) \end{array}\] \pause \begin{remark} All sorts of $X_\UU$ are reached by some $\Cstr$ \end{remark} \end{frame} \begin{frame}{Constructing $F_3$} \renewcommand\arraystretch{1.8} \[ \begin{array}{l|l} X : \mathbf{\BB_3} & Y = F_3(X): \mathbf{\CC_3}\\\hline X_\UU : \Set\quad X_\El : \Set^{X_\UU} &\\ \Cstr_\Con : X_\UU & Y_\Con = \uncover<2->{\ensuremath{X_\El(\Cstr^X_\Con)}}\\ \Cstr_\Ty : H_2 F_{1} (X,\Cstr_\Con) \to X_\UU & Y_\Ty(\Gamma) = \uncover<3->{\ensuremath{X_\El(\Cstr^X_\Ty(\Gamma))}}\\ \Cstr_\Tm : H_3 F_{2} (X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU & Y_\Tm(\Delta,A) = \uncover<4->{\ensuremath{X_\El(\Cstr^X_\Tm(\Delta,A))}} \\ \end{array} \] \pause[5] \begin{remark} Each object of $Y$ are associated by $X_\El$ to some $\Cstr$ \end{remark} \end{frame} \begin{frame}{Adjunction $F \vdash G$} \[ \Hom_{\BB_3}\left(G_3Y,X\right) \simeq \Hom_{\CC_3}\left(Y,F_3X\right) \] \pause \begin{remark} All sorts of $G_3Y$ are reached by some $\Cstr$ of $G_3Y$ \uncover<3>{\ding{220}$\Hom_{\BB_3}\left(G_3Y,X\right)$ transforms constructible into constructible} Each object of $F_3X$ are associated by $X_\El$ to some $\Cstr$ of $X$ \uncover<3>{\ding{220}$\Hom_{\CC_3}\left(Y,F_3X\right)$ transforms constructible into constructible} \end{remark} \end{frame} \section{The complete proof \& Discoveries} \begin{frame}{Structure of the global proof} \begin{itemize} \item Categories $\CC_i$ \quad $\BB_i$ \item Functors $F_i : \BB_i \to \CC_i : G_i$ \item Adjunction $F_i \vdash G_i$ \item Forgetful functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$ \item Operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ \quad $\inj_1^i : X \to X \tl^i Y$ \quad $\inj_2^i : Y \to R_0^i(X \tl^i Y)$ \item Coreflection $F_iG_i \cong \Id_{\CC_i}$ \item Isomorphism $F_i\inj_1^i$ \item Isomorphism $(R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$ \end{itemize} \end{frame} \begin{frame}{Fibration of $\CC_i$} \end{frame} \begin{frame}{$S_i$ from syntax} \end{frame} \section{Conclusion} \begin{frame}{Conclusion} \end{frame} \begin{frame}{Future work} \begin{itemize} \item Complete GAT (term constructors + equalities) \item Proof Assistant Formalization \item $S_i$ non-direct \end{itemize} \end{frame} \begin{frame} \begin{center} \Large Thank you for your attention \end{center} \end{frame} \end{document}