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This commit is contained in:
Samy Avrillon 2024-09-03 22:58:12 +02:00
parent b8ed28e8e4
commit 3955bd89f2
Signed by: Mysaa
GPG Key ID: 0220AC4A3D6A328B
2 changed files with 42 additions and 18 deletions
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54
Report/M2Diapo.tex
View File

@ -336,6 +336,7 @@
\end{frame}
\begin{frame}{Constructing $G_3$}
\vspace{-2ex}
\begin{columns}
\begin{column}{.5\textwidth}
\begin{center}
@ -356,30 +357,41 @@
\end{array}\]
}
\pause
\[\begin{array}{ccccccc}
\renewcommand{\arraystretch}{0.5}
\[\begin{array}{ccccccccc}
X_\UU & = &
\{\star\} & \uplus &
Y_\Con & \uplus&
\displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta) \\
X_\El(\inj_1(\star)) & = &
Y_\Con &&&&\\
X_\El(\inj_2(\Gamma)) & = &
&&Y_\Ty(\Gamma) &&\\
X_\El(\inj_3(\Delta,A)) & = &
&&&& Y_\Tm(\Delta,A)\\
\displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta) &
\xrightarrow{X_\El}&\Set\\
\pause
&& \tikzmark{1}{\star} &&&&&&\tikzmark{2}{Y_\Con}\\
X_\El & &
&&\tikzmark{3}{\Gamma}&&&&\tikzmark{4}{Y_\Ty(\Gamma)}\\
&&&&&& \tikzmark{5}{(\Delta,A)}&& \tikzmark{6}{Y_\Tm(\Delta,A)}\\
\end{array}\]
\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}
\[\begin{array}{lcl}
\XCon_X &=& \inj_1(\star)\\
\XTy_X(\Gamma) &=& \inj_2(\Gamma) \\
\XTm_X(\Delta,A) &=& \inj_3(\Delta,A)
\XCon_X &=& \star \in \{\star\}\\
\XTy_X(\Gamma) &=& \Gamma \in Y_\Con \\
\XTm_X(\Delta,A) &=& (\Delta,A) \in \displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta)
\end{array}\]
\end{column}
\pause
\vspace{-1.5ex}
\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 $F_3$}
@ -405,7 +417,7 @@
\pause[5]
\vspace{-.5ex}
\begin{tcolorbox}
Each object of $Y$ are associated by $X_\El$ to some constructor
The preimage by $X_\El$ of each object of $Y$ is a sort of $X_\UU$ constructed by some constructor
\end{tcolorbox}
\end{frame}
@ -417,13 +429,18 @@
\pause
\begin{remark}
All sorts of $G_3Y$ are reached by some constructor of $G_3Y$
Morphisms of $\BB_3$ and $\CC_3$ both respect constructors
\pause
\uncover<3>{\ding{220}$\Hom_{\BB_3}\left(G_3Y,X\right)$ transforms constructible into constructible}
All sorts of $G_3Y_\UU$ are in the image of some constructor $(\XCon_{G_3Y},\XTy_{G_3Y},\XTm_{G_3Y})$
Each object of $F_3X$ are associated by $X_\El$ to some constructor of $X$
\ding{220}$\Hom_{\BB_3}\left(G_3Y,X\right)$ morphisms are only defined on constructors
\uncover<3>{\ding{220}$\Hom_{\CC_3}\left(Y,F_3X\right)$ transforms constructible into constructible}
\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}
@ -431,7 +448,7 @@
\begin{frame}{Conclusion}
\begin{center}
\hspace{2ex}$\CC$ \hspace{3.5cm} $\BB$
\hspace{9.2ex}$\CC$ \hspace{3.7cm} $\BB$
\vspace{.5cm}
\includesvg[scale=.4]{graphs/diagrammeFG.svg}
@ -510,3 +527,4 @@

6
Report/headerDiapo.tex
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@ -191,3 +191,9 @@
\tableofcontents[currentsection,hideothersubsections,sections=\value{section}]
\end{frame}
}
\newcommand{\tikzmark}[2]{\tikz[remember picture] \node (#1) {$#2$};}
\makeatletter
\newcommand{\mathleft}{\@fleqntrue\@mathmargin0pt}
\newcommand{\mathcenter}{\@fleqnfalse}
\makeatother