Quick fixes
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@ -2,6 +2,7 @@
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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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@ -20,8 +21,8 @@
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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 dependent \textbf{type theory}
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\item[\ding{220}] Enables to create sorts, object of those sorts, equalities between those objects
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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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@ -35,10 +36,10 @@
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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}{l}
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$A : \Set$ \\
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$B : \Set$ \\\hline
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$\operatorname{exec} : A \to B$
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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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@ -52,18 +53,19 @@
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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}{.6\textwidth}
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\begin{column}{.7\textwidth}
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\renewcommand\arraystretch{1.5}
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\begin{tabular}{l}
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$A : \Set$ \\
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$B : \Set$ \\\hline
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$\operatorname{exec} : A \to B$ \\
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$\operatorname{invexec} : B \to A$\\\hline
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$\operatorname{isol} : (x : A) \to \operatorname{invexec}(\operatorname{exec}\;x) = x$\\
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$\operatorname{isor} : (y : B) \to \operatorname{exec}(\operatorname{invexec}\;y) = y$\\
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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}{.4\textwidth}
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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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@ -80,12 +82,12 @@
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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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$\Hom : \Obj \to \Obj \to \Set$ \\\hline
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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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$\operatorname{comp} : (A\;B\;C:\Obj) \to \Hom\;B\;C \to\Hom\;A\;B \to \Hom\;A\;C$ \\\hline
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$\operatorname{idl}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \operatorname{comp} (\id\;B) \sigma = \sigma$\\
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$\operatorname{idr}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \operatorname{comp} \sigma (\id\;A) = \sigma$\\
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$\operatorname{comp-trans}: \dots$\\
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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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@ -105,7 +107,7 @@
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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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$\Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set$ \\\hline
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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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@ -129,12 +131,17 @@
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{2-sortification}
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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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\ding{229} Simplify a study of GATs
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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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@ -142,7 +149,7 @@
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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 $x : \El\;o$ & $\leftrightarrow$ & $x : \mathcal{O}$ \pause\\\hline
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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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@ -171,7 +178,7 @@
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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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{\large How to state this fact ?}
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\ding{229} Semantical proof
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@ -185,10 +192,11 @@
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\section{One Example}
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\begin{frame}{Constructing the categories}
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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 = sorts + \sout{constructors} + \sout{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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@ -202,7 +210,7 @@
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&
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$\begin{array}{l}
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\mathcal{O} : \Set\\
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\El : \mathcal{O} \to \Set \\\hline
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\El : \mathcal{O} \to \Set \\\hdashline
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\underline{\Con} : \mathcal{O} \\
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\underline{\Ty} : \El\;\underline{\Con} \to \mathcal{O} \\
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\underline{\Tm} : (\Gamma : \El\;\underline{\Con}) \to \El(\underline{\Ty}\;\Gamma) \to \mathcal{O}
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@ -222,27 +230,27 @@
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\end{center}
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\end{frame}
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\begin{frame}{Category of Models \& Generalization}
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\begin{frame}{Categories of Models}{of the original GAT}
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\begin{tabular}{lcp{0.5\textwidth}}
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$\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]$}
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\only<1>{$\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]$}
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\only<1>{$\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<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<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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@ -254,7 +262,7 @@
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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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\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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@ -277,55 +285,70 @@
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\end{frame}
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\begin{frame}{Adding transformed sort declarations}
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\begin{frame}{Category of Models}{of the transformed GAT}
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\begin{center}
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\color{blue}{
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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{tabular}}
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\end{center}
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\begin{center}
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\renewcommand\arraystretch{1.4}
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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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$(X_\UU,X_\El)$ \\
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$\boxed{\Con : \mathcal{O}}$ & $\Cstr_\Con :$ &
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$\boxed{\Con : \mathcal{O}}$ & $\XCon_X :$ &
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\only<1>{$X_\UU$}
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\only<2>{$H_1F_0(X) \to X_\UU$} \\
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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(\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>{\rule[6ex]{0pt}{0pt}\renewcommand\arraystretch{0.9}$\begin{array}{c}\left(\Delta \in X_\El(\Cstr_\Con)\right) \to\\ \left(A \in X_\El(\Cstr_\Ty(\Delta))\right) \to\\ X_\UU\end{array}$}
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\only<2>{$H_3F_2(X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU$}
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$\boxed{\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}}$ & $\XTy_X :$&
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\only<1>{$\left(\Gamma \in X_\El(\XCon_X)\right) \to X_\UU $}
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\only<2>{$H_2F_1(X,\XCon_X) \to X_\UU$} \\
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$\boxed{\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}}$ &
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$\XTm_X : $ &
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\only<1>{\rule[6ex]{0pt}{0pt}\renewcommand\arraystretch{0.9}$\begin{array}{c}\left(\Delta \in X_\El(\XCon_X)\right) \to\\ \left(A \in X_\El(\XTy_X(\Delta))\right) \to\\ X_\UU\end{array}$}
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\only<2>{$H_3F_2(X,\XCon_X,\XTy_X) \to X_\UU$}
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\end{tabular}
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\end{center}
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\pause
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\begin{center}
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\includesvg[scale=.4]{graphs/diagrammeFGmini.svg}
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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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\begin{frame}{Constructing $F$ and $G$}
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\begin{center}
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\includesvg[scale=.4]{graphs/diagrammeFGmini.svg}
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\end{center}
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\[
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\begin{array}{l|l}
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Y : \mathbf{\CC_3} & X : \mathbf{\BB_3} \\\hline
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& X_\UU : \Set \quad X_\El : \Set^{X_\UU}\\
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Y_\Con : \Set & \Cstr_\Con : X_\UU \\
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Y_\Con : \Set & \XCon_X : X_\UU \\
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\left(Y_\Ty(\Gamma)\right)_{\Gamma \in Y_\Con} &
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\Cstr_\Ty : H_2 F_{1} (X,\Cstr_\Con) \to X_\UU \\
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\XTy_X : H_2 F_{1} (X,\XCon_X) \to X_\UU \\
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\left(\left(Y_\Tm(\Delta,A)\right)_{A \in Y_\Ty(\Delta)}\right)_{\Delta \in Y_\Con} &
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\Cstr_\Tm : H_3 F_{2} (X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU \\
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\XTm_X : H_3 F_{2} (X,\XCon_X,\XTy_X) \to X_\UU \\
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\end{array}
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\]
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\begin{center}
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\begin{tabular}{ll}
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$X_\UU$:& 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}{Constructing $G_3$}
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\begin{columns}
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\begin{column}{.5\textwidth}
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\begin{center}
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\includesvg[scale=.4]{graphs/diagrammeGmini.svg}
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\end{center}
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\end{column}
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\begin{column}{.5\textwidth}
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\[
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Y = \left(Y_\Con,Y_\Ty,Y_\Tm\right)
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\]
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\end{column}
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\end{columns}
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\vspace{.5ex}
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\only<1>{
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\[\begin{array}{ccl}
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X_\UU & = & \text{«sorts»}\\
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@ -335,8 +358,8 @@
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\pause
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\[\begin{array}{ccccccc}
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X_\UU & = &
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1 & \oplus &
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Y_\Con & \oplus&
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\{\star\} & \uplus &
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Y_\Con & \uplus&
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\displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta) \\
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X_\El(\inj_1(\star)) & = &
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Y_\Con &&&&\\
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@ -348,37 +371,42 @@
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\end{array}\]
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\pause
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\[\begin{array}{lcl}
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\Cstr_\Con &=& \inj_1(\star)\\
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\Cstr_\Ty(\Gamma) &=& \inj_2(\Gamma) \\
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\Cstr_\Tm(\Delta,A) &=& \inj_3(\Delta,A)
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\XCon_X &=& \inj_1(\star)\\
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\XTy_X(\Gamma) &=& \inj_2(\Gamma) \\
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\XTm_X(\Delta,A) &=& \inj_3(\Delta,A)
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\end{array}\]
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\pause
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\begin{remark}
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All sorts of $X_\UU$ are reached by some $\Cstr$
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\end{remark}
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\vspace{-1.5ex}
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\begin{tcolorbox}
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All sorts of $X_\UU$ are in the image of some constructor $(\XCon_X,\XTy_X,\XTm_X)$
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\end{tcolorbox}
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\end{frame}
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\begin{frame}{Constructing $F_3$}
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\renewcommand\arraystretch{1.8}
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\begin{center}
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\includesvg[scale=.4]{graphs/diagrammeFmini.svg}
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\end{center}
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\vspace{-1ex}
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\renewcommand\arraystretch{1.5}
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\[
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\begin{array}{l|l}
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X : \mathbf{\BB_3} & Y = F_3(X): \mathbf{\CC_3}\\\hline
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X_\UU : \Set\quad X_\El : \Set^{X_\UU} &\\
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\Cstr_\Con : X_\UU & Y_\Con =
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\uncover<2->{\ensuremath{X_\El(\Cstr^X_\Con)}}\\
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\Cstr_\Ty : H_2 F_{1} (X,\Cstr_\Con) \to X_\UU &
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\XCon_X : X_\UU & Y_\Con =
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\uncover<2->{\ensuremath{X_\El(\XCon_X)}}\\
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\XTy_X : H_2 F_{1} (X,\XCon_X) \to X_\UU &
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Y_\Ty(\Gamma) =
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\uncover<3->{\ensuremath{X_\El(\Cstr^X_\Ty(\Gamma))}}\\
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\Cstr_\Tm : H_3 F_{2} (X,\Cstr_\Con,\Cstr_\Ty) \to X_\UU &
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\uncover<3->{\ensuremath{X_\El(\XTy_X(\Gamma))}}\\
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\XTm_X : H_3 F_{2} (X,\XCon_X,\XTy_X) \to X_\UU &
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Y_\Tm(\Delta,A) =
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\uncover<4->{\ensuremath{X_\El(\Cstr^X_\Tm(\Delta,A))}} \\
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\uncover<4->{\ensuremath{X_\El(\XTm_X(\Delta,A))}} \\
|
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\]
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\pause[5]
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Each object of $Y$ are associated by $X_\El$ to some $\Cstr$
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\vspace{-.5ex}
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@ -389,11 +417,11 @@
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\pause
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\uncover<3>{\ding{220}$\Hom_{\BB_3}\left(G_3Y,X\right)$ transforms constructible into constructible}
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Each object of $F_3X$ are associated by $X_\El$ to some $\Cstr$ of $X$
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Each object of $F_3X$ are associated by $X_\El$ to some constructor of $X$
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\uncover<3>{\ding{220}$\Hom_{\CC_3}\left(Y,F_3X\right)$ transforms constructible into constructible}
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After Width: | Height: | Size: 18 KiB |
@ -31,6 +31,7 @@
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\usepackage[backend=biber,style=numeric]{biblatex}
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\usepackage{array}
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\usepackage{arydshln}
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\usepackage[lighttt]{lmodern}
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\usetikzlibrary{shapes.geometric,positioning,backgrounds}
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@ -84,6 +85,9 @@
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\newcommand\Con{{\ensuremath{\operatorname{Con}}}}
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\newcommand\Ty{{\ensuremath{\operatorname{Ty}}}}
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\newcommand\Tm{{\ensuremath{\operatorname{Tm}}}}
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\newcommand\XCon{{\ensuremath{\operatorname{\mathbf{Con}}}}}
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\newcommand\XTy{{\ensuremath{\operatorname{\mathbf{Ty}}}}}
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\newcommand\XTm{{\ensuremath{\operatorname{\mathbf{Tm}}}}}
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\newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}
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\newcommand\Rtsc{{\ensuremath{\operatorname{\mathcal{R}tsc}}}}
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@ -186,4 +190,4 @@
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\pdfpcnote{#1}
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\tableofcontents[currentsection,hideothersubsections,sections=\value{section}]
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\end{frame}
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}
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}
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|
||||
Loading…
x
Reference in New Issue
Block a user