More paragraphs

Report/M2Report.tex

@@ -152,13 +152,10 @@
 		We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one.
 		The categories $\CC_i$ are described as in Fiore's paper \cite{Fiore2008}, and the categories $\BB_i$ are constructed atop of the category $\TSet$ with a method inspired by the category of models described by Altenkirch et al. \cite{Altenkirch2018}.
 		
-		The overall construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
+		The overall recursive construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
 		
 		\begin{center}
 			% YADE DIAGRAM G1.json
-			% GENERATED LATEX
-			\input{graphs/G1.tex}
-			% END OF GENERATED LATEX
 		\end{center}
 		
 		The first step of our recursion is the trivial adjunction $\lambda . \star \vdash \lambda . 1$ between the categories $\BB_0 = \TSet$ and $\CC_0 = 1$.
@@ -167,11 +164,14 @@
 		
 		We construct the category $\CC_i$ as the following pair:
 		\[
-			\CC_i = (X : \CC_{i-1}) \times \Set^{\Hom(O_i,X)} \qquad\text{(this is a dependent coproduct)}
+			\CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/\Hom(O_i,X)\right)
 		\]
-		where $O_i$ is a specific object of the category $\CC_{i-1}$, such that $\Hom(O_i,X)$ is the set of parameters for the construction of the new sort.
+		where for any category $\mathcal{C}$ and $X$ an object of $\mathcal{C}$, $(\mathcal{C}/X)$ it the slice category (or over category) of arrows pointing out of $X$ (its objects $(Y,f)$ are arrows $f : X \to Y$ and its morphisms are morphisms creating commutative triangles).\inlinetodo{Assez clair ?} \inlinetodo{On ne voit pas que $(\Set/A(X)) \cong \Set^{A(X)}$}
 		
-		For example, for our type theory example, we first have
+		and where $O_i$ is a specific object of the category $\CC_{i-1}$, such that $\Hom(O_i,X)$ is the set of parameters for the construction of the new sort.
+		\todo{Comment indiquer que la paire est dépendante ?}
+		
+		I will give now an example of those $O_i$ objects for our type theory example. We begin with
 		\[
 			O_1 = \star \in \operatorname{Obj}(\CC_0) = \operatorname{Obj}(1)
 		\]
@@ -291,13 +291,15 @@
 		W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}O_i,\dash)}(X,Y), \widetilde{\inj_2} \right)
 		\]
 		
-		Where $\Hbar_A(X,Y)$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ with 
-		\[\begin{array}{c}
-			\Hbar_A(X,Y)_\UU = A(X)\\
-			\Hbar_A(X,Y)_\El = Y
-		\end{array}\]
+		Where $\Hbar_A$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ defined as
+		\[
+		\Hbar_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)}
+		\]
+		The morphisms are translated as-is.
 		
-		\todo{Réference de comment on crée le foncteur, pourquoi c'en est un, si c'est utile ...}
+		\begin{remark}
+			This functor can be constructed as a lax colimit seeing elements of $A(X)/\Set$ as lax cocones over the diagram $\left[1 \xrightarrow{A(X)} \Set\right]$ in $\Cat$, and seeing elements of $\TSet$ as lax cocones over the diagram with no arrow $\left[\Set \quad \Set\right]$. \inlinetodo{Vérifier ça}
+		\end{remark}
 		
 		With $\widetilde{\inj_2}$ being defined by \inlinetodo{Changer les noms des hypothèses H3' et H1r}
 		\[
@@ -753,3 +755,7 @@
 
 
 
+
+
+
+

Report/header.tex

@@ -109,6 +109,7 @@
 \newcommand\Ty{{\ensuremath{\operatorname{Ty}}}}
 \newcommand\Tm{{\ensuremath{\operatorname{Tm}}}}
 \newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}
+\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\Hom{{\ensuremath{\operatorname{\mathcal{H}om}}}}