Correction d'erreurs

Report/M2Diapo.tex

@@ -37,21 +37,51 @@
 	
 	\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
-				$\operatorname{exec} : A \to B$ \\
+				$\operatorname{exec} : A \to B$
-				\pause
-				$\operatorname{invexec} : B \to A$\\\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)
+			\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}
 				$\Obj : \Set$ \\
@@ -62,11 +92,20 @@
 				$\operatorname{idr}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \operatorname{comp} \sigma (\id\;A) = \sigma$\\
 				$\operatorname{comp-trans}: \dots$\\
 			\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}
 				$\Con : \Set$ \\
@@ -78,6 +117,20 @@
 				$\operatorname{app} : (\Gamma : \Con) \to (A\;B :\Ty\;\Gamma) \to$\\
 				\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}
@@ -91,12 +144,13 @@
 	\begin{frame}{2-sortification of the Set Function GAT}
 		\begin{tcolorbox}
 			\renewcommand\arraystretch{1.5}
-			\begin{tabular}{ll}
+			\begin{tabular}{llrcl}
-				$\mathcal{O} : \Set$ & \color{RoyalBlue} \text{sorts} \\
+				\pause
-				$\El : \mathcal{O} \to \Set$ & \color{RoyalBlue}\text{objects of that sort} \pause\\\hline
+				$\mathcal{O} : \Set$ & \color{RoyalBlue} \text{sorts} & $o:\mathcal{O}$ &$\leftrightarrow$&$o$ is a sort \\
-				$A : \mathcal{O}$ &\\
+				$\El : \mathcal{O} \to \Set$ & \color{RoyalBlue}\text{objects of that sort} & \qquad $x : \El\;o$ & $\leftrightarrow$ &  $x : \mathcal{O}$ \pause\\\hline
-				$B : \mathcal{O}$ &\\
+				$A : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$A : \Set$}&&\\
-				$\operatorname{exec} : \El\;A \to \El\;B$ &
+				$B : \mathcal{O}$ &&\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}
@@ -121,6 +175,7 @@
 		
 		\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
@@ -129,15 +184,52 @@
 			\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\inlinetodo{Ptet trop pointu}
+		\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}}
-			$\boxed{\bullet}$ &	$\CC_0 :=$ &
+			$\boxed{()}$ &	$\CC_0 :=$ &
 			$\one$ \\
 			$\boxed{\Con : \Set}$ & $\CC_1 := $&
 			\only<1>{$\left[X_\Con\right]$}
@@ -179,7 +271,6 @@
 		
 		\[
 			\BB_0 := \left(X_\UU : \Set, X_\El : \Set^{X_\UU}\right)
-			{\color{PineGreen}\ensuremath{\simeq \left(X_\UU : \Set, X_\El : \Set/X_\UU\right)}}
 		\]
 		\pause
 		\begin{center}
@@ -200,9 +291,10 @@
 			\end{tabular}
 		\end{center}
 		\begin{center}
+		\renewcommand\arraystretch{1.4}
 		\begin{tabular}{lrp{0.4\textwidth}}
 			$\boxed{\mathcal{O} : \Set\quad\El : \mathcal{O} \to \Set}$ &	$\BB_0 =$ &
-			$X : \TSet$ \\
+			$(X_\UU,X_\El)$ \\
 			$\boxed{\Con : \mathcal{O}}$ & $\Cstr_\Con :$ &
 			\only<1>{$X_\UU$}
 			\only<2>{$H_1F_0(X) \to X_\UU$} \\
@@ -210,7 +302,7 @@
 			\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>{$\begin{array}{c}\left(\Delta \in X_\El(\Cstr_\Con)\right) \to\\ \left(A \in X_\El(\Cstr_\Ty(\Delta))\right) \to\\ X_\UU\end{array}$}
+			\only<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}
@@ -254,9 +346,9 @@
 			X_\El(\inj_1(\star)) & = &
 			Y_\Con &&&&\\
 			X_\El(\inj_2(\Gamma)) & = &
-			&&\displaystyle\coprod_{\Gamma \in Y_\Con}Y_\Ty(\Gamma) &&\\
+			&&Y_\Ty(\Gamma) &&\\
 			X_\El(\inj_3(\Delta,A)) & = &
-			&&&& \displaystyle\coprod_{\Delta \in Y_\Con}\coprod_{A \in Y_\Ty(\Delta)}Y_\Ty(\Delta,A)\\
+			&&&& Y_\Tm(\Delta,A)\\
 			
 		\end{array}\]
 		\pause
@@ -288,7 +380,7 @@
 			\uncover<4->{\ensuremath{X_\El(\Cstr^X_\Tm(\Delta,A))}} \\
 		\end{array}
 		\]
-		\pause[4]
+		\pause[5]
 		\begin{remark}
 			Each object of $Y$ are associated by $X_\El$ to some $\Cstr$
 		\end{remark}
@@ -304,30 +396,28 @@
 		\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}
-		\pause
-		\begin{property}
-			$FG \cong \Id$
-		\end{property}
 	\end{frame}
 	
 	\section{The complete proof \& Discoveries}
 	
-	\begin{frame}{Structure of the proof}
+	\begin{frame}{Structure of the global proof}
 		\begin{itemize}
-			\item $\CC_i$
+			\item Categories $\CC_i$ \quad $\BB_i$
-			\item $\BB_i$
+			\item Functors $F_i : \BB_i \to \CC_i : G_i$
-			\item $R_{i-1}^i : \BB_i \to \BB_{i-1}$
+			\item Adjunction $F_i \vdash G_i$
-					$R_0^i : \BB_i \to \BB_0$
+			\item Forgetful functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$
-			\item $\tl^i : \BB_i \times \BB_0 \to \BB_i$
+			\item Operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ \quad
-					$\inj_1^i : X \to X \tl^i Y$
+					$\inj_1^i : X \to X \tl^i Y$ \quad
 					$\inj_2^i : Y \to R_0^i(X \tl^i Y)$
-			\item $F_i : \BB_i \to \CC_i : G_i$
+			\item Coreflection $F_iG_i \cong \Id_{\CC_i}$
-			\item $F_i \vdash G_i$
+			\item Isomorphism $F_i\inj_1^i$
-			\item $F_iG_i \cong \Id_{\CC_i}$
+			\item Isomorphism $(R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$
-			$F_i\inj_1^i$
-			\item $(R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$
 		\end{itemize}
 	\end{frame}
 	
@@ -353,6 +443,12 @@
 		\end{itemize}
 	\end{frame}
 	
+	\begin{frame}
+		\begin{center}
+			\Large Thank you for your attention
+		\end{center}
+	\end{frame}
+	
 \end{document} 
 
 

Report/headerDiapo.tex

@@ -27,6 +27,7 @@
 \usepackage{environ}
 \usepackage{pifont}
 \usepackage{transparent}
+\usepackage{ulem}
 \usepackage[backend=biber,style=numeric]{biblatex}
 \usepackage{hyperref}
 \usepackage{array}