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Samy Avrillon 2024-09-03 11:35:58 +02:00
parent c36520f1a3
commit a7df1d8369
Signed by: Mysaa
GPG Key ID: 0220AC4A3D6A328B
2 changed files with 52 additions and 32 deletions
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80
Report/M2Diapo.tex
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@ -17,11 +17,6 @@
\maketitle
\end{frame}
\section*{Table of contents}
\begin{frame}{Plan of the presentation}
\tableofcontents[hidesubsections]
\end{frame}
\section{GATs and 2-sortification}
\begin{frame}{What is a GAT ?}
\begin{itemize}
@ -404,35 +399,15 @@
\end{remark}
\end{frame}
\section{The complete proof \& Discoveries}
\begin{frame}{Structure of the global proof}
\begin{itemize}
\item Categories $\CC_i$ \quad $\BB_i$
\item Functors $F_i : \BB_i \to \CC_i : G_i$
\item Adjunction $F_i \vdash G_i$
\item Forgetful functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$
\item Operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ \quad
$\inj_1^i : X \to X \tl^i Y$ \quad
$\inj_2^i : Y \to R_0^i(X \tl^i Y)$
\item Coreflection $F_iG_i \cong \Id_{\CC_i}$
\item Isomorphism $F_i\inj_1^i$
\item Isomorphism $(R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$
\end{itemize}
\end{frame}
\begin{frame}{Fibration of $\CC_i$}
\end{frame}
\begin{frame}{$S_i$ from syntax}
\end{frame}
\section{Conclusion}
\begin{frame}{Conclusion}
\begin{center}
\hspace{2ex}$\CC$ \hspace{3.5cm} $\BB$
\vspace{.5cm}
\includesvg[scale=.4]{graphs/diagrammeFG.svg}
\end{center}
\end{frame}
\begin{frame}{Future work}
@ -449,6 +424,51 @@
\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}

2
Report/headerDiapo.tex
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@ -171,7 +171,7 @@
\hypersetup{pdfpagemode=FullScreen}
% Transition en fade-in par défaut
\addtobeamertemplate{background canvas}{\transfade[duration=0.4]}{}
%\addtobeamertemplate{background canvas}{\transfade[duration=0.4]}{}
\addtobeamertemplate{frametitle}{
\let\insertframetitle\insertsubsectionhead}{}