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.build/
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.build/
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graphs/*.tex
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graphs/*.tex
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svg-inkscape
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% !TeX spellcheck = en_US
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\DocumentMetadata{}
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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{GATs and 2-sortification}
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\begin{frame}{What is a GAT ?}
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\begin{itemize}
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\item A \textbf{list} of declarations
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\item[\ding{220}] Enables to declare sorts, objects of those sorts and equalities between those objects
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\item A \textbf{syntactic object}
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\item[\ding{220}] Type judgements are defined with induction rules
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\item Describes \textbf{models}
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\item[\ding{220}] 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 a function in Set}
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\begin{tcolorbox}
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\begin{columns}
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\begin{column}{.6\textwidth}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{ll}
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$A : \Set$&\multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{sorts}$}\\
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$B : \Set$&\\\hdashline
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$\operatorname{exec} : A \to B$&$\}\text{constructors}$
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\end{tabular}
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\end{column}
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\begin{column}{.4\textwidth}
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Models:
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triples (A,B,f)
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\end{column}
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\end{columns}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{A GAT for an bijective function in Set}
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\begin{tcolorbox}
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\begin{columns}
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\begin{column}{.7\textwidth}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{ll}
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$A : \Set$ & \multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{sorts}$}\\
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$B : \Set$ &\\\hdashline
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$\operatorname{exec} : A \to B$ &\multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{constructors}$}\\
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$\operatorname{invexec} : B \to A$&\\\hdashline
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$\operatorname{isol} : (x : A) \to \operatorname{invexec}(\operatorname{exec}\;x) = x$&
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\multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{equalities}$}\\
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$\operatorname{isor} : (y : B) \to \operatorname{exec}(\operatorname{invexec}\;y) = y$&\\
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\end{tabular}
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\end{column}
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\begin{column}{.3\textwidth}
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Models:
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triples (A,B,f)
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s.t. f is bijective
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\end{column}
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\end{columns}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{A GAT for a small category}
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\begin{tcolorbox}
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\begin{columns}
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\begin{column}{0.8\textwidth}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{l}
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$\Obj : \Set$ \\
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$\Hom : \Obj \to \Obj \to \Set$ \\\hdashline
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$\id : (A : \Obj) \to \Hom\;A\;A$ \\
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$\circ : (A\;B\;C:\Obj) \to \Hom\;B\;C \to\Hom\;A\;B \to \Hom\;A\;C$ \\\hdashline
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$\operatorname{idl}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \circ\;(\id\;B)\;\sigma = \sigma$\\
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$\operatorname{idr}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \circ\;\sigma\;(\id\;A) = \sigma$\\
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$\operatorname{\circ-trans}: \dots$\\
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\end{tabular}
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\end{column}
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\begin{column}{0.2\textwidth}
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Models:
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small categories
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\end{column}
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\end{columns}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{A GAT for Type Theory}
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\begin{tcolorbox}
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\begin{columns}
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\begin{column}{0.7\textwidth}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{l}
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$\Con : \Set$ \\
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$\Ty : \Con \to \Set$ \\
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$\Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set$ \\\hdashline
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$\operatorname{empty}: \Con$ \\
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$\operatorname{ext}: (\Gamma:\Con)\to(A:\Ty\;\Gamma)\to\Con$ \\
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$\operatorname{implies} : (\Gamma:\Con) \to \Ty\;\Gamma \to \Ty\;\Gamma \to \Ty\;\Gamma$ \\
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$\operatorname{app} : (\Gamma : \Con) \to (A\;B :\Ty\;\Gamma) \to$\\
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\qquad$\Tm\;\Gamma\;(\operatorname{implies}\;A\;B)\to\Tm\;\Gamma\; A \to \Tm\;\Gamma\; B$
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\end{tabular}
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\end{column}
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\begin{column}{0.3\textwidth}
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Models : Triples
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$(X_\Con,X_\Ty,X_\Tm)$
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with constructors
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$\operatorname{empty} \in X_\Con$
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\qquad\vdots
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\end{column}
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\end{columns}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{Subject of my internship}{2-sortification of GATs}
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\begin{center}
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{\large Transform a GAT into a GAT with only two sorts}
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\end{center}
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\vspace{3ex}
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\ding{229} Process observed since at least 2021
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\ding{229} Never formally studied
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\ding{229} Studying all GATs by only studying GATs with two sorts ?
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\end{frame}
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\begin{frame}{2-sortification of the Set Function GAT}
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\begin{tcolorbox}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{llrcl}
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\pause
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$\mathcal{O} : \Set$ & \color{RoyalBlue} \text{sorts} & $o:\mathcal{O}$ &$\leftrightarrow$&$o$ is a sort \\
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$\El : \mathcal{O} \to \Set$ & \color{RoyalBlue}\text{objects of that sort} & \qquad $x : \El\;o$ & $\leftrightarrow$ & $x : o$ \pause\\\hline
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$A : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$A : \Set$}&&\\
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$B : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$B : \Set$}&&\\
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$\operatorname{exec} : \El\;A \to \El\;B$&&\color{teal}$\operatorname{exec} : A \to B$&&
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\end{tabular}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{2-sortification of Type Theory GAT}
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\begin{tcolorbox}
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\renewcommand\arraystretch{1.4}
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|
\begin{tabular}{l}
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$\mathcal{O} : \Set$ \\
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$\El : \mathcal{O} \to \Set$ \pause\\\hline
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$\Con : \mathcal{O}$ \\
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$\Ty : \El\;\Con \to \mathcal{O}$ \\
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$\Tm : (\Gamma : \El\;\Con) \to \El\;(\Ty\;\Gamma) \to \mathcal{O}$ \\
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$\operatorname{empty}: \El\;\Con$ \\
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$\operatorname{ext}: (\Gamma:\El\;\Con)\to(A:\El\;(\Ty\;\Gamma))\to\El\;\Con$ \\
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$\operatorname{implies} : (\Gamma:\El\;\Con) \to \El\;(\Ty\;\Gamma) \to \El\;(\Ty\;\Gamma) \to \El\;(\Ty\;\Gamma)$
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\end{tabular}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{Goal of the internship}{}
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{\large Is this transformation correct ?}
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\ding{229} Can one study all GATs by studying only GATs with two sorts
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\vspace{1ex}
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{\large How to state this fact ?}
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\ding{229} Semantical proof
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\begin{center}
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\includesvg[scale=.4]{graphs/diagrammeFG.svg}
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\end{center}
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\ding{229} This adjunction proves that one can make the initial model of any GAT from the initial model of the transformed GAT
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\end{frame}
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\section{One Example}
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\begin{frame}{Categories of models}
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\begin{center}
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\renewcommand{\ULthickness}{2.4pt}
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\only<1>{GAT = sorts + constructors + equalities}
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\only<2>{GAT = \textbf{sorts} + \sout{constructors} + \sout{equalities}}
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\end{center}
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\pause[2]
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\begin{center}
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\begin{tabular}{|l|l|}
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\hline
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$\begin{array}{l}
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\Con : \Set \\
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\Ty : \Con \to \Set \\
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\Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set
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\end{array}$
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&
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$\begin{array}{l}
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\mathcal{O} : \Set\\
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\El : \mathcal{O} \to \Set \\\hdashline
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\underline{\Con} : \mathcal{O} \\
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\underline{\Ty} : \El\;\underline{\Con} \to \mathcal{O} \\
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\underline{\Tm} : (\Gamma : \El\;\underline{\Con}) \to \El(\underline{\Ty}\;\Gamma) \to \mathcal{O}
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\end{array}$\\\hline\rule{0pt}{1.0\normalbaselineskip}
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\only<2>{$\CC \hookrightarrow (\Con,\Ty,\Tm)$}
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\only<3->{$\CC_0 \hookrightarrow ()$}
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&
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\only<2>{$\BB \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty},\underline{\Tm})$}
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\only<3->{$\BB_0 \hookrightarrow (\mathcal{O},\El)$}\\
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\uncover<3->{$\CC_1 \hookrightarrow (\Con)$} &
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\uncover<3->{$\BB_1 \hookrightarrow (\mathcal{O},\El,\underline{\Con})$}\\
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\uncover<3->{$\CC_2 \hookrightarrow (\Con,\Ty)$} &
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\uncover<3->{$\BB_2 \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty})$}\\
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\uncover<3->{$\CC_3 \hookrightarrow (\Con,\Ty,\Tm)$} &
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\uncover<3->{$\BB_3 \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty},\underline{\Tm})$}\\\hline
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\end{tabular}
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\end{center}
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\end{frame}
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\begin{frame}{Categories of Models}{of the original GAT}
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\begin{tabular}{lcp{0.5\textwidth}}
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$\boxed{()}$ & $\CC_0 :=$ &
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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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|
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\pause[4]
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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 the 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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\[
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\BB_0 := \left(X_\UU : \Set, X_\El : \Set^{X_\UU}\right)
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\]
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\pause
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\begin{center}
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\begin{tabular}{ll}
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$X_\UU$:& Sorts \\
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$X_\El(o)$:& Objects of sort $o$\\
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\end{tabular}
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\end{center}
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\end{frame}
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\begin{frame}{Category of Models}{of the transformed GAT}
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||||||
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\begin{center}
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||||||
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\color{blue}{
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||||||
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\begin{tabular}{ll}
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||||||
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$X_\UU$:& Sorts \\
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||||||
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$X_\El(o)$:& Objects of sort $o$\\
|
||||||
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\end{tabular}}
|
||||||
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\end{center}
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||||||
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\begin{center}
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||||||
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\renewcommand\arraystretch{1.4}
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||||||
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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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||||||
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$(X_\UU,X_\El)$ \\
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$\boxed{\Con : \mathcal{O}}$ & $\XCon_X :$ &
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\only<1>{$X_\UU$}
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\only<2>{$H_1F_0(X) \to X_\UU$} \\
|
||||||
|
$\boxed{\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}}$ & $\XTy_X :$&
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||||||
|
\only<1>{$\left(\Gamma \in X_\El(\XCon_X)\right) \to X_\UU $}
|
||||||
|
\only<2>{$H_2F_1(X,\XCon_X) \to X_\UU$} \\
|
||||||
|
$\boxed{\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}}$ &
|
||||||
|
$\XTm_X : $ &
|
||||||
|
\only<1>{\rule[6ex]{0pt}{0pt}\renewcommand\arraystretch{0.9}$\begin{array}{c}\left(\Delta \in X_\El(\XCon_X)\right) \to\\ \left(A \in X_\El(\XTy_X(\Delta))\right) \to\\ X_\UU\end{array}$}
|
||||||
|
\only<2>{$H_3F_2(X,\XCon_X,\XTy_X) \to X_\UU$}
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
\pause
|
||||||
|
\begin{center}
|
||||||
|
\includesvg[scale=.4]{graphs/diagrammeFGmini.svg}
|
||||||
|
\end{center}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
\begin{frame}{Constructing $F$ and $G$}
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\includesvg[scale=.4]{graphs/diagrammeFGmini.svg}
|
||||||
|
\end{center}
|
||||||
|
\[
|
||||||
|
\begin{array}{l|l}
|
||||||
|
Y : \mathbf{\CC_3} & X : \mathbf{\BB_3} \\\hline
|
||||||
|
& X_\UU : \Set \quad X_\El : \Set^{X_\UU}\\
|
||||||
|
Y_\Con : \Set & \XCon_X : X_\UU \\
|
||||||
|
\left(Y_\Ty(\Gamma)\right)_{\Gamma \in Y_\Con} &
|
||||||
|
\XTy_X : H_2 F_{1} (X,\XCon_X) \to X_\UU \\
|
||||||
|
\left(\left(Y_\Tm(\Delta,A)\right)_{A \in Y_\Ty(\Delta)}\right)_{\Delta \in Y_\Con} &
|
||||||
|
\XTm_X : H_3 F_{2} (X,\XCon_X,\XTy_X) \to X_\UU \\
|
||||||
|
\end{array}
|
||||||
|
\]
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
\begin{frame}{Constructing $G_3$}
|
||||||
|
\vspace{-2ex}
|
||||||
|
\begin{columns}
|
||||||
|
\begin{column}{.5\textwidth}
|
||||||
|
\begin{center}
|
||||||
|
\includesvg[scale=.4]{graphs/diagrammeGmini.svg}
|
||||||
|
\end{center}
|
||||||
|
\end{column}
|
||||||
|
\begin{column}{.5\textwidth}
|
||||||
|
\[
|
||||||
|
Y = \left(Y_\Con,Y_\Ty,Y_\Tm\right)
|
||||||
|
\]
|
||||||
|
\end{column}
|
||||||
|
\end{columns}
|
||||||
|
\vspace{.5ex}
|
||||||
|
\only<1>{
|
||||||
|
\[\begin{array}{ccl}
|
||||||
|
X_\UU & = & \text{«sorts»}\\
|
||||||
|
X_\El(o) & = & \text{«objects of sort $o$»}
|
||||||
|
\end{array}\]
|
||||||
|
}
|
||||||
|
\pause
|
||||||
|
\renewcommand{\arraystretch}{0.5}
|
||||||
|
\[\begin{array}{ccccccccc}
|
||||||
|
X_\UU & = &
|
||||||
|
\{\star\} & \uplus &
|
||||||
|
Y_\Con & \uplus&
|
||||||
|
\displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta) &
|
||||||
|
\xrightarrow{X_\El}&\Set\\
|
||||||
|
\pause
|
||||||
|
&& \tikzmark{1}{\star} &&&&&&\tikzmark{2}{Y_\Con}\\
|
||||||
|
X_\El & &
|
||||||
|
&&\tikzmark{3}{\Gamma}&&&&\tikzmark{4}{Y_\Ty(\Gamma)}\\
|
||||||
|
&&&&&& \tikzmark{5}{(\Delta,A)}&& \tikzmark{6}{Y_\Tm(\Delta,A)}\\
|
||||||
|
|
||||||
|
\end{array}\]
|
||||||
|
\begin{tikzpicture}[overlay,remember picture]
|
||||||
|
\draw[|->] (1) to (2);
|
||||||
|
\draw[|->] (3) to (4);
|
||||||
|
\draw[|->] (5) to (6);
|
||||||
|
\end{tikzpicture}
|
||||||
|
\pause
|
||||||
|
\begin{columns}
|
||||||
|
\begin{column}{.5\textwidth}
|
||||||
|
\[\begin{array}{lcl}
|
||||||
|
\XCon_X &=& \star \in \{\star\}\\
|
||||||
|
\XTy_X(\Gamma) &=& \Gamma \in Y_\Con \\
|
||||||
|
\XTm_X(\Delta,A) &=& (\Delta,A) \in \displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta)
|
||||||
|
\end{array}\]
|
||||||
|
\end{column}
|
||||||
|
\pause
|
||||||
|
\begin{column}{.5\textwidth}
|
||||||
|
\begin{tcolorbox}
|
||||||
|
All sorts of $X_\UU$ are in the image of some constructor $(\XCon_X,\XTy_X,\XTm_X)$
|
||||||
|
\end{tcolorbox}
|
||||||
|
\end{column}
|
||||||
|
\end{columns}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
\begin{frame}{Constructing $F_3$}
|
||||||
|
\begin{center}
|
||||||
|
\includesvg[scale=.4]{graphs/diagrammeFmini.svg}
|
||||||
|
\end{center}
|
||||||
|
\vspace{-1ex}
|
||||||
|
\renewcommand\arraystretch{1.5}
|
||||||
|
\[
|
||||||
|
\begin{array}{l|l}
|
||||||
|
X : \mathbf{\BB_3} & Y = F_3(X): \mathbf{\CC_3}\\\hline
|
||||||
|
X_\UU : \Set\quad X_\El : \Set^{X_\UU} &\\
|
||||||
|
\XCon_X : X_\UU & Y_\Con =
|
||||||
|
\uncover<2->{\ensuremath{X_\El(\XCon_X)}}\\
|
||||||
|
\XTy_X : H_2 F_{1} (X,\XCon_X) \to X_\UU &
|
||||||
|
Y_\Ty(\Gamma) =
|
||||||
|
\uncover<3->{\ensuremath{X_\El(\XTy_X(\Gamma))}}\\
|
||||||
|
\XTm_X : H_3 F_{2} (X,\XCon_X,\XTy_X) \to X_\UU &
|
||||||
|
Y_\Tm(\Delta,A) =
|
||||||
|
\uncover<4->{\ensuremath{X_\El(\XTm_X(\Delta,A))}} \\
|
||||||
|
\end{array}
|
||||||
|
\]
|
||||||
|
\pause[5]
|
||||||
|
\vspace{-.5ex}
|
||||||
|
\begin{tcolorbox}
|
||||||
|
The preimage by $X_\El$ of each object of $Y$ is a sort of $X_\UU$ constructed by some constructor
|
||||||
|
\end{tcolorbox}
|
||||||
|
|
||||||
|
\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}
|
||||||
|
Morphisms of $\BB_3$ and $\CC_3$ both respect constructors
|
||||||
|
\pause
|
||||||
|
|
||||||
|
All sorts of $G_3Y_\UU$ are in the image of some constructor $(\XCon_{G_3Y},\XTy_{G_3Y},\XTm_{G_3Y})$
|
||||||
|
|
||||||
|
\ding{220}$\Hom_{\BB_3}\left(G_3Y,X\right)$ morphisms are only defined on constructors
|
||||||
|
|
||||||
|
\pause
|
||||||
|
|
||||||
|
The preimage by $X_\El$ of each object of $F_3X$ is a sort of $X_\UU$ constructed by some constructor
|
||||||
|
|
||||||
|
\ding{220}$\Hom_{\CC_3}\left(Y,F_3X\right)$ morphisms only send to constructible terms
|
||||||
|
\end{remark}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
\section{Conclusion}
|
||||||
|
|
||||||
|
\begin{frame}{Conclusion}
|
||||||
|
\begin{center}
|
||||||
|
\hspace{9.2ex}$\CC$ \hspace{3.7cm} $\BB$
|
||||||
|
\vspace{.5cm}
|
||||||
|
|
||||||
|
\includesvg[scale=.4]{graphs/diagrammeFG.svg}
|
||||||
|
\end{center}
|
||||||
|
\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}
|
||||||
|
|
||||||
|
\appendix
|
||||||
|
|
||||||
|
\begin{frame}
|
||||||
|
\[\begin{array}{lcl}
|
||||||
|
F_3G_3(Y)_\Con &=& G_3(Y)_p^{-1}(\{\Cstr^{G_3(Y)}_\Con\})\\
|
||||||
|
&=& G_3(Y)_p^{-1}(\{\inj_1 \star\}) \\
|
||||||
|
&=& Y_\Con
|
||||||
|
\end{array}\]
|
||||||
|
and
|
||||||
|
\[\begin{array}{lcl}
|
||||||
|
F_3G_3(Y)_\Ty(\Gamma) &=& G_3(Y)_p^{-1}(\{\Cstr^{G_3(Y)}_\Ty(\Gamma)\})\\
|
||||||
|
&=& G_3(Y)_p^{-1}(\{\inj_2 \Gamma\}) \\
|
||||||
|
&=& \operatorname{proj}_1^{-1}(\Gamma) \\
|
||||||
|
&=& \left\{(\Gamma',A) \in \coprod_{\Gamma' \in Y_\Con}Y_\Ty(\Gamma') \middle| \Gamma' = \Gamma\right\}\\
|
||||||
|
&\simeq& Y_\Ty(\Gamma)
|
||||||
|
\end{array}\]
|
||||||
|
and finally, with the same method, we get that
|
||||||
|
\[
|
||||||
|
F_3G_3(Y)_\Tm(\Delta,A) \simeq Y_\Tm(\Delta,A)
|
||||||
|
\]
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
\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}
|
||||||
|
|
||||||
|
\end{document}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
+713
-443
File diff suppressed because it is too large
Load Diff
@@ -0,0 +1,71 @@
|
|||||||
|
# M2 Presentation
|
||||||
|
|
||||||
|
### Introduction slide
|
||||||
|
Notes on my internship
|
||||||
|
|
||||||
|
### Plan of the presentation
|
||||||
|
First present what is the subject of my study, and what i tried to achieve.
|
||||||
|
|
||||||
|
The whole proof is too formal and complex: Informally prove one specific example.
|
||||||
|
|
||||||
|
Then, i will present the «structure» of the global proof, and show some things discovered along the way
|
||||||
|
|
||||||
|
## GAT and 2-sortification
|
||||||
|
### What is a GAT
|
||||||
|
|
||||||
|
### A GAT for a function in set
|
||||||
|
A GAT can be composed of three kinds of lines:
|
||||||
|
- Sort declarations
|
||||||
|
- Objects costructors
|
||||||
|
A model is the data of a function from a set to another set i.e. a triple the data of two sets, and a «mapping» between them
|
||||||
|
We can spice up this GAT: Gat of isomorphic functions.
|
||||||
|
- Equalities
|
||||||
|
|
||||||
|
### A GAT for a category
|
||||||
|
Another example of GATs:
|
||||||
|
A category is the data of ...... with ...... such that ......
|
||||||
|
|
||||||
|
### A GAT for Type Theory
|
||||||
|
Last example that of type theory
|
||||||
|
-> a lot to add, contructors, equalities ..., just for the example.
|
||||||
|
|
||||||
|
### 2-sortification
|
||||||
|
Known process
|
||||||
|
Used by Sestini
|
||||||
|
Can simplify a study of GATs to only GATs of two sorts
|
||||||
|
|
||||||
|
### 2-sortification of the Set function GAT
|
||||||
|
Sort specification is always the same. O will describe «sorts» and El will describe «objects of that sort»
|
||||||
|
|
||||||
|
Then, we replace Set occuneces by O and we prefix everything else by El
|
||||||
|
|
||||||
|
### 2-sortification of the Type Theoyr GAT
|
||||||
|
|
||||||
|
### Goal of the internship
|
||||||
|
|
||||||
|
## One Example
|
||||||
|
### Category of Models & Generalization
|
||||||
|
|
||||||
|
### Category of models of the Transformed GAT
|
||||||
|
|
||||||
|
### Adding transformed sort declarations
|
||||||
|
|
||||||
|
### Constructing the functors
|
||||||
|
|
||||||
|
### Contsructing G3
|
||||||
|
|
||||||
|
### Constructing F3
|
||||||
|
|
||||||
|
### Adjunction FG
|
||||||
|
|
||||||
|
## The complete proof & Discoveries
|
||||||
|
### Structure of the proof
|
||||||
|
|
||||||
|
### Fibration of C
|
||||||
|
|
||||||
|
### S from syntax
|
||||||
|
|
||||||
|
## Conclusion
|
||||||
|
### Conclusion
|
||||||
|
|
||||||
|
### Future work
|
||||||
@@ -0,0 +1 @@
|
|||||||
|
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\CC{{\\ensuremath{\\mathcal{C}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":2,"label":{"kind":"normal","label":"F_0","style":{"alignment":"right","bend":0.10000000000000003,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":3,"label":{"kind":"normal","label":"G_0","style":{"alignment":"right","bend":0.10000000000000003,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":0},{"from":2,"id":4,"label":{"kind":"adjunction","label":"\\vdash","style":{"alignment":"over","bend":0,"color":"black","dashed":false,"head":"none","kind":"none","position":0.5,"tail":"none"},"zindex":0},"to":3}],"nodes":[{"id":0,"label":{"isMath":true,"label":"\\TSet = \\BB_0","pos":[300,97],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"\\CC_0 = \\one","pos":[572,97],"zindex":0}}],"sizeGrid":200,"title":"1"}]},"version":12}
|
||||||
@@ -1 +1 @@
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id="path332" />
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<path
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</g>
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||||||
|
</g>
|
||||||
|
</svg>
|
||||||
|
After Width: | Height: | Size: 18 KiB |
+37
-9
@@ -42,11 +42,10 @@
|
|||||||
\usepackage{newunicodechar}
|
\usepackage{newunicodechar}
|
||||||
\usepackage{txfonts}
|
\usepackage{txfonts}
|
||||||
\usepackage{yade}
|
\usepackage{yade}
|
||||||
|
\usepackage{environ}
|
||||||
\usepackage[backend=biber,style=numeric]{biblatex}
|
\usepackage[backend=biber,style=numeric]{biblatex}
|
||||||
\usepackage{hyperref}
|
\usepackage{hyperref}
|
||||||
|
|
||||||
\usepackage[textheight=0.75\paperheight]{geometry}
|
|
||||||
|
|
||||||
\usepackage[lighttt]{lmodern}
|
\usepackage[lighttt]{lmodern}
|
||||||
\usetikzlibrary{shapes.geometric,positioning,backgrounds}
|
\usetikzlibrary{shapes.geometric,positioning,backgrounds}
|
||||||
|
|
||||||
@@ -68,8 +67,6 @@
|
|||||||
|
|
||||||
|
|
||||||
% Création des environnement globaux
|
% Création des environnement globaux
|
||||||
\newtheorem{theorem}{Theorem}
|
|
||||||
\newtheorem{definition}{Definition}
|
|
||||||
\newtheorem{property}{Property}
|
\newtheorem{property}{Property}
|
||||||
\newtheorem{remark}{Remark}
|
\newtheorem{remark}{Remark}
|
||||||
|
|
||||||
@@ -126,13 +123,13 @@
|
|||||||
\node (El) at (0,1) {\El};
|
\node (El) at (0,1) {\El};
|
||||||
\draw[->] (El) -- node[anchor=east] {\ensuremath{\scriptstyle p}} (U);
|
\draw[->] (El) -- node[anchor=east] {\ensuremath{\scriptstyle p}} (U);
|
||||||
|
|
||||||
\node (XU) at (1,0) {\ensuremath{#3}};
|
\node (XU) at (1.5,0) {\ensuremath{#3}};
|
||||||
\node (XEl) at (1,1) {\ensuremath{#1}};
|
\node (XEl) at (1.5,1) {\ensuremath{#1}};
|
||||||
\draw[->] (XEl) -- node[anchor=west] {\ensuremath{\scriptstyle #2}} (XU);
|
\draw[->] (XEl) -- node[anchor=west] {\ensuremath{\scriptstyle #2}} (XU);
|
||||||
|
|
||||||
\draw[|->] (.3,0) -- (.7,0);
|
\draw[|->] (.3,0) -- (1,0);
|
||||||
\draw[|->] (.3,0.5) -- (.7,0.5);
|
\draw[|->] (.3,0.5) -- (1,0.5);
|
||||||
\draw[|->] (.3,1) -- (.7,1);
|
\draw[|->] (.3,1) -- (1,1);
|
||||||
|
|
||||||
\coordinate (base) at (.5,.4);
|
\coordinate (base) at (.5,.4);
|
||||||
\end{tikzpicture}
|
\end{tikzpicture}
|
||||||
@@ -146,11 +143,18 @@
|
|||||||
\path[->] (0) edge["${\scriptstyle #2}$", pos=0.5, fore, black,=>, ] (1);
|
\path[->] (0) edge["${\scriptstyle #2}$", pos=0.5, fore, black,=>, ] (1);
|
||||||
\end{tikzpicture}
|
\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\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\todo[1]{\begin{tcolorbox}[colback=red!20!white,colframe=red!85!black,boxrule=4pt] #1 \end{tcolorbox}}
|
||||||
\newcommand\inlinetodo[1]{\colorbox{orange}{#1}}
|
\newcommand\inlinetodo[1]{\colorbox{orange}{#1}}
|
||||||
|
|
||||||
|
|
||||||
% Création des environnements spécifiques au document
|
% Création des environnements spécifiques au document
|
||||||
|
|
||||||
|
|
||||||
@@ -161,4 +165,28 @@
|
|||||||
\titleformat{\subparagraph}[runin]{\normalfont\normalsize\bfseries}{}{0em}{\subparaghaphboxedcontent}
|
\titleformat{\subparagraph}[runin]{\normalfont\normalsize\bfseries}{}{0em}{\subparaghaphboxedcontent}
|
||||||
\titlespacing*{\subparagraph}{0pt}{3.25ex plus 1ex minus .2ex}{0.5em}
|
\titlespacing*{\subparagraph}{0pt}{3.25ex plus 1ex minus .2ex}{0.5em}
|
||||||
|
|
||||||
|
% Fixing Yade green
|
||||||
|
\definecolor{green}{RGB}{11,102,35}
|
||||||
|
|
||||||
|
% Environ for scaling tikz pictures
|
||||||
|
\makeatletter
|
||||||
|
\newsavebox{\measure@tikzpicture}
|
||||||
|
\NewEnviron{scaletikzpicturetowidth}[1]{%
|
||||||
|
\def\tikz@width{#1}%
|
||||||
|
\def\tikzscale{1}\begin{lrbox}{\measure@tikzpicture}%
|
||||||
|
\BODY
|
||||||
|
\end{lrbox}%
|
||||||
|
\pgfmathparse{#1/\wd\measure@tikzpicture}%
|
||||||
|
\edef\tikzscale{\pgfmathresult}%
|
||||||
|
\BODY
|
||||||
|
}
|
||||||
|
\makeatother
|
||||||
|
|
||||||
|
\usepackage{fancyhdr}
|
||||||
|
\makeatletter
|
||||||
|
\makeatother
|
||||||
|
\fancyfoot[C]{}
|
||||||
|
\fancyfoot[R]{\textbf{\thepage / \pageref{LastReportPage}}}
|
||||||
|
\pagestyle{fancy}
|
||||||
|
|
||||||
\addbibresource{Bilibibio.bib}
|
\addbibresource{Bilibibio.bib}
|
||||||
@@ -0,0 +1,199 @@
|
|||||||
|
% Loading packages
|
||||||
|
\usepackage{fontspec}
|
||||||
|
\usepackage[utf8]{inputenc}
|
||||||
|
\usepackage{amsmath}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{bbm}
|
||||||
|
\usepackage{stmaryrd}
|
||||||
|
\usepackage[english]{babel}
|
||||||
|
\usepackage{csquotes}
|
||||||
|
\usepackage{listings}
|
||||||
|
\usepackage{lstautogobble}
|
||||||
|
\usepackage{svg}
|
||||||
|
\usepackage{tikz}
|
||||||
|
\usepackage{multirow}
|
||||||
|
\usepackage{multicol}
|
||||||
|
\usepackage{tcolorbox}
|
||||||
|
\usepackage{mdframed}
|
||||||
|
\usepackage{proof}
|
||||||
|
\usepackage{xparse}
|
||||||
|
\usepackage{cprotect}
|
||||||
|
\usepackage{xpatch}
|
||||||
|
\usepackage{amsmath}
|
||||||
|
\usepackage{amsfonts}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{txfonts}
|
||||||
|
\usepackage{yade}
|
||||||
|
\usepackage{environ}
|
||||||
|
\usepackage{pifont}
|
||||||
|
\usepackage{transparent}
|
||||||
|
\usepackage{ulem}
|
||||||
|
\usepackage[backend=biber,style=numeric]{biblatex}
|
||||||
|
\usepackage{hyperref}
|
||||||
|
\usepackage{array}
|
||||||
|
\usepackage{arydshln}
|
||||||
|
|
||||||
|
\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\XCon{{\ensuremath{\operatorname{\mathbf{Con}}}}}
|
||||||
|
\newcommand\XTy{{\ensuremath{\operatorname{\mathbf{Ty}}}}}
|
||||||
|
\newcommand\XTm{{\ensuremath{\operatorname{\mathbf{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\Obj{{\ensuremath{\operatorname{\mathcal{O}bj}}}}
|
||||||
|
\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]}{}
|
||||||
|
|
||||||
|
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@@ -1,56 +0,0 @@
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% !TeX spellcheck = fr_FR
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\documentclass[11pt]{article}
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\usepackage[francais]{babel}
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\usepackage{csquotes}
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\usepackage{xcolor}
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% \setlength{\parskip}{0.3\baselineskip}
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\newcommand\question[1]{\textbf{\textcolor{orange}{#1}}}
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%\newcommand\question[1]{}
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\begin{document}
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\title{Sémantique catégorique de la réduction des GATs en GATs à deux sortes}
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\author{Samy Avrillon, encadré par Ambroise Lafont\\
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\small Équipe PARTOUT, Laboratoire d'Informatique de l'École Polytechnique (LIX)}
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\maketitle
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\pagestyle{empty} %
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\thispagestyle{empty}
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\subsection*{Le contexte général}
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Les théories algébriques généralisées (ou GAT) sont des objects syntaxiques introduits en 1986 par Cartmell qui permettent la description de structures algébriques, que l'on peut voir comme une généralisation des types inductifs de la théorie des types. Par exemple, on peut décrire les modèles d'une théorie des types à l'aide d'un GAT.
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Un GAT est constitué d'une liste de \enquote{sortes} décrivant les ensembles, généralement suivie d'une liste de \enquote{constructeurs}.
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\subsection*{Le problème étudié}
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Il existe un processus qui permet de transformer n'importe quel GAT en un GAT avec seulement deux sortes. Cette transformation a notamment été exploitée par Filippo Sestini dans sa thèse, afin de restreindre son sujet d'étude. Cependant, il n'a pas été prouvé que cette transformation ne réduisait pas la généralité de sa thèse.
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\subsection*{La contribution proposée}
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Durant ce stage, j'ai chercher à formaliser sémantiquement cette transformation. C'est à dire que je me suis intéressé directement aux catégories des modèles sans m'attacher à l'aspect syntaxique des GATs. Je me suis restreint à l'étude des \enquote{spécification de sortes}, c'est à dire aux GATs constitués uniquement d'une liste de sortes, sans constructeurs. J'ai commencé par appliquer la transformation sur des exemples simples, afin d'identifier des propriétés qui la justifient, le tout dans un cadre catégorique. Ensuite, j'ai énoncé et prouvé formellement ces propriétés dans leur plus grande généralité.
|
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||||||
|
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||||||
\subsection*{Les arguments en faveur de sa validité}
|
|
||||||
|
|
||||||
La preuve a été faite sémantiquement en essayant à chaque étape de généraliser les objets utilisés au maximum. Les constructions des objets sont basées sur des papiers déjà publiés, qui sont cités dans ce rapport.
|
|
||||||
|
|
||||||
La construction proposées et les propriétés établies prouvent également la conjecture établie par Philippo Sestini dans sa thèse.
|
|
||||||
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|
||||||
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||||||
\subsection*{Le bilan et les perspectives}
|
|
||||||
|
|
||||||
Nous avons formalisé la transformation sémantiquement, ce qui permet de gagner en généralité et de s'abstraire de l'aspect syntaxique des GATs.
|
|
||||||
Cette tranformation a été prouvée, et permet notamment de justifier une restriction d'une étude aux seuls GATs à deux sortes.
|
|
||||||
|
|
||||||
Une prochaine étape serait de rajouter les constructeurs de termes à notre formalisation, en décrivant leur transformation et en adaptant les propriétés prouvées pour le cas des spécifications de sortes.
|
|
||||||
|
|
||||||
Nous avons également de quoi réfléchir à une généralisation des GATs qui décriraient par exemple des sortes définies l'une mutuellement contenue dans l'autre, et d'autres sortes plus \enquote{exotiques}. Il y a du travail à les étudier, et à observer l'effet de la transformation sur ces objets.
|
|
||||||
|
|
||||||
\end{document}
|
|
||||||
@@ -1,5 +1,5 @@
|
|||||||
{
|
{
|
||||||
"watchedFile": "M2Report.tex",
|
"watchedFile": "M2Diapo.tex",
|
||||||
"baseDir": "graphs",
|
"baseDir": "graphs",
|
||||||
"externalOutput": true,
|
"externalOutput": true,
|
||||||
"preambleFile": "yade-preamble.tex"
|
"preambleFile": "yade-preamble.tex"
|
||||||
|
|||||||
Reference in New Issue
Block a user