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Ajout d'une partie exemples

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% END OF GENERATED LATEX
\end{center}
\section{Constructing the coreflection}
\section{Example}
\paragraph{Structure of the proof}
Before making the formal proof, we will construct and explain the objects on two examples.
\subsection{Structure of the proof}
We will formalize the transformation for \emph{sort specifications} (i.e. lists of sort declarations).
This proof will be an induction on the number of sorts taken into account. At each step, we will add a new sort declaration, represented by a functor $H_i$ described later (\autoref{sec:constructingCategory}).
At the $i$-th recursion step, we will build the following objects :
\begin{itemize}
\setlength\itemsep{-1ex}
\item The category of models of the GAT $\CC_i$
\item The category of models of the transformed GAT $\BB_i$
\item A forgetful functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$.
We will formalize the transformation for \emph{sort specifications} (i.e. lists of sort declarations).
Those functors compose themselves into a functor $R_0^i : \BB_i \to \BB_0$:
\[
R_0^i := R^1_0 \circ \dots \circ R_{i-1}^i
\]
\item An operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ along with a morphism $\inj_\tl^i : X \to X \tl^i Y$ for every $X,Y$ in $\BB_i \times \BB_0$, such that the canonical morphism $\en_{i-1}^i : (R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$ is an isomorphism. This operation follows a specific universal property: For every morphisms $g : X \to Z$ and $h : Y \to R_0^iZ$, there is an unique morphism $\{g;h\}$ such that the two following diagrams commute:
\label{sec:tlUniversalProperty}
This proof is an induction on the number of sorts taken into account. At each step, we will add a new sort declaration, represented by a functor $H_i$ described later (\autoref{sec:constructingCategory}).
\begin{center}
% YADE DIAGRAM TlUniversal.json
% GENERATED LATEX
\input{graphs/TlUniversal.tex}
% END OF GENERATED LATEX
\end{center}
At the $i$-th recursion step, we will build the following objects :
\begin{itemize}
\setlength\itemsep{-1ex}
\item The category of models of the GAT $\CC_i$
\item The category of models of the transformed GAT $\BB_i$
\item A forgetful functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$ that creates limits. In our type theory example, $R_1^2$ is a functor from the category of models of $(\mathcal{O},\El,\Con,\Ty,\Tm)$ to the category of models of $(\mathcal{O},\El,\Con,\Ty)$.
where $\en_0^i$ is the following composition:
\[\en_0^i := R_0^{i-1} \en_{i-1}^i \circ \dots \circ R_0^1\en_1^2 \circ \en_0^1 : (R_0^i X) \oplus Y = (R_0^i X) \tl^0 Y \to R_0^i (X \tl^i Y)\]
And where $\inj_2$ is the second injector of $\oplus$, the coproduct of the category $\BB_0 = \TSet$. We will define $\tl^0$ to be $\oplus$, so the equality above holds.
The operator is also defined on morphisms, such that for any morphism $g : X \to X'$ in $\BB_i$ and $h : Y \to Y'$ in $\BB_0$, there is a morphism $g \tl^i h : X \tl^i Y \to X' \tl^i Y'$ in $\BB_i$.
\item A functor $F_i : \BB_i \to \CC_i$
\item A functor $G_i : \CC_i \to \BB_i$
\item An adjunction between them $F_i \vdash G_i$
\item A proof that $F_iG_i \cong \Id_{\CC_i}$ (i.e. $F_i \vdash G_i$ make up a coreflection)
\item A proof that $F_i\inj_\tl^i$ is an isomorphism. The inverse isomorphism will be denoted as $(F_i\inj_\tl^i)^{-1}$
\end{itemize}
\subsection{The empty sort specification}
Our proof will we an induction on the number of sorts, and so the first step will be creating the objects for the empty sort specification.
Therefore we will build the following objects
\paragraph{Category of models of the empty sort specification}
The category of models of the empty sort specification will simply be $\one$, the category with only one object $\star$ and one trivial morphism (i.e. the terminal category of $\Cat$). This is because the empty sort specification only has one model. We will denote this category as $\CC_0$.
Those functors compose themselves into a functor $R_0^i : \BB_i \to \BB_0$, that also creates limits:
\[
R_0^i := R^1_0 \circ \dots \circ R_{i-1}^i
\]
\item An operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ along with a morphism $\inj_\tl^i : X \to X \tl^i Y$ for every $X,Y$ in $\BB_i \times \BB_0$, such that the canonical morphism $\en_{i-1}^i : (R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$ is an isomorphism. This operation follows a specific universal property: For every morphisms $g : X \to Z$ and $h : Y \to R_0^iZ$, there is an unique morphism $\{g;h\}$ such that the two following diagrams commute:
\label{sec:tlUniversalProperty}
\paragraph{Category of models of the transformed GAT}
The transformed GAT for the empty sort specification is the sort specification $(\mathcal{O},\El)$ (no constructors as there are no sort in the initial sort specification):
\vspace{.5ex}
\begin{center}
\renewcommand\arraystretch{1}
\begin{tabular}{|l|}
\hline
$\UU : \Set$ \\
$\El : \mathcal{O} \to \Set$ \\
\hline
\end{tabular}
\end{center}
\vspace{.5ex}
The usual way of defining the category of models of this sort specification is by taking the category of families of sets $\FamSet$, whose objects are pairs of a set $X$ and a family of sets indexed by $X$ : $(Y_x)_{x\in X}$. However, we will rather use another category : $\TSet$, the category of presheaves over the category $\TT$, the category with two object and one non-trivial arrow between them. The objects and morphism of $\TT$ are described below:
\begin{center}
% YADE DIAGRAM TlUniversal.json
% YADE DIAGRAM CategoryTT.json
% GENERATED LATEX
\input{graphs/TlUniversal.tex}
\input{graphs/CategoryTT.tex}
% END OF GENERATED LATEX
\end{center}
where $\en_0^i$ is the following composition:
\[\en_0^i := R_0^{i-1} \en_{i-1}^i \circ \dots \circ R_0^1\en_1^2 \circ \en_0^1 : (R_0^i X) \oplus Y = (R_0^i X) \tl^0 Y \to R_0^i (X \tl^i Y)\]
This category is equivalent to the catgegory $\FamSet$ with an equivalence we will see later\inlinetodo{Insérer la référence}
And where $\inj_2$ is the second injector of $\oplus$, the coproduct of the category $\BB_0 = \TSet$. We will define $\tl^0$ to be $\oplus$, so the equality above holds.
With this formalisation, a model of this sort specification is a functor $X : \TT \to \Set$, such that
\begin{itemize}
\item $X_\UU$ is the set of the \enquote{sort objects}
\item For each sort object $\Gamma \in X_\UU$, the set of objects corresponding to the sort object is $X_p^{-1}(\{\Gamma\}) \subset X_\El$
\end{itemize}
The operator is also defined on morphisms, such that for any morphism $g : X \to X'$ in $\BB_i$ and $h : Y \to Y'$ in $\BB_0$, there is a morphism $g \tl^i h : X \tl^i Y \to X' \tl^i Y'$ in $\BB_i$.
\item A functor $F_i : \BB_i \to \CC_i$
\item A functor $G_i : \CC_i \to \BB_i$
\item An adjunction between them $F_i \vdash G_i$
\item A proof that $F_iG_i \cong \Id_{\CC_i}$ (i.e. $F_i \vdash G_i$ make up a coreflection)
\item A proof that $F_i\inj_\tl^i$ is an isomorphism. The inverse isomorphism will be denoted as $(F_i\inj_\tl^i)^{-1}$
\end{itemize}
We will denote this category of models of the empty sort specification as $\BB_0$.
We notice that the category $\BB_0$ has a coproduct that we will denote as $\tl^0$. The two injectors of this coproduct will be denoted as $\inj_1^0$ and $\inj_2^0$.
\paragraph{Functors}
We want to show that there is a reflection between those two categories $\BB_0$ and $\CC_0$ as follows:
\begin{center}
% YADE DIAGRAM AdjunctionLevel0.json
% GENERATED LATEX
\input{graphs/AdjunctionLevel0.tex}
% END OF GENERATED LATEX
\end{center}
$F_0 : \TSet \to \one$ is simply the terminal functor of $\Cat$, i.e. the functor sending all objects to $\star$, the only object of $\one$, and all morphisms to $\id_\star$.
$G_0 : \one \to \TSet$ is the functor that sends the only object of $\one$ to the initial object of $\TSet$: $0_\TSet$ (such that $G_0$ preserves colimits)
We can check that those two functors create an adjunction, as
\[
\Hom(G_0 \star,X) = \Hom(0_\TSet,X) \cong \one \cong \Hom(\star,F_0 X)
\]
We also have that they create a coreflection, as $F_0G_0$ goes from $\one$ to $one$, and so $F_0G_0 = \Id_\one$ as $\one$ is terminal in $\Cat$. So $F_0G_0$ is an isofunctor.
\subsection{Type theory example}
We will take as a first example the sort specification of type theory presented in the introduction. The sort specification and its transformation are as follows:
\vspace{1ex}
\begin{center}
\renewcommand\arraystretch{1}
\begin{tabular}{|l|l|}
\hline
& $\UU : \Set$ \\
& $\El : \mathcal{O} \to \Set$ \\
$\Con : \Set$ & $\Con : \mathcal{O}$\\
$\Ty : (\Gamma : \Con) \to \Set$ & $\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}$\\
$\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set$ & $\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}$ \\
\hline
\end{tabular}
\end{center}
\vspace{1ex}
We will build the following objects, step by step, adding one sort at a time:
\begin{center}
% YADE DIAGRAM TypeTheoryConstruction.json
% GENERATED LATEX
\input{graphs/TypeTheoryConstruction.tex}
% END OF GENERATED LATEX
\end{center}
Where the categories $\CC_i$ are the categories of models of the sort specification limited to the $i$ first sort declarations, and $\BB_i$ are the categories of models of the transformed GAT limited to the $i$ first sort declarations. For example, $\CC_2$ is the category of models of the two first sorts of the sort specification ($\Con$,$\Ty$). $\BB_0$,$\CC_0$,$F_0$, and $G_0$ have been defined in the last subsection.
The $R^i_{i-1}$ functors between the categories are forgetful functors that forgets about the added sorts.
I will only give here the explanation of functors $F_3$ and $G_3$, omitting the intermediate functors $F_1$,$G_1$,$F_2$,$G_2$. Their construction is given in the complete proof.
\paragraph{Category of models}
We have seen in the introduction how was a model of the type theory sort specification. We will rebuild that inductively, adding one sort at a time.
In order to construct those categories in a generic way, we will define functors $H_X : \CC_{X-1} \to \Set$ that will describe the \enquote{parameters} of the sort declaration.
\[
\boxed{\Con : \Set}
\]
We begin with the following functor, corresponding to the sort declaration above
\[
H_\Con(\star) = 1 \in \Set
\]
This functor means that $\Con$ takes no parameter i.e. there is only \emph{one} way of constructing a $\Con$.
We build $\CC_1$ to be a pair of
\begin{itemize}
\item An object $X_0$ of $\CC_0$
\item A family of sets $X_\Con$ indexed by $H_\Con(x) = 1$
\end{itemize}
As there is only one object in $\CC_0$ and that a family of sets indexed by $1$ is simply a set, we have that this category $\CC_1$ is isomorphic to $\Set$ (i.e. the category of sets).
This is as expected: a model $X$ consists of a set $X_\Con$.
\[
\boxed{\Ty : (\Gamma : \Con) \to \Set}
\]
Then, we take the functor $H_2(X) = X_\Con$, corresponding to the sort declaration above. This functor means that $\Ty$ takes one parameter of type $\Con$ i.e. There is exactly one $\Ty$ sort for every $\Con$ there is in the model.
Again, we build $\CC_\Ty$ to be a pair of
\begin{itemize}
\item A model $X_\Con$ from $\CC_1$
\item A family of sets $X_\Ty$ indexed by $H_\Ty(X_\Con) = X_\Con$
\end{itemize}
This category is, as expected, the category $\FamSet$ of families of sets i.e. objects are pairs of a set $X_\Con$ and of a family indexed by $X_\Con$: $(X_\Ty(\Gamma))_{\Gamma\in X_\Con}$.
\[
\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}
\]
The functor $\CC_2 \to \Set$ we will use here is defined as follows:
\[
H_\Tm(X_\Con,X_\Ty) = \coprod_{\Delta \in \Con}X_\Ty(\Delta) \in \Set
\]
$\coprod$ denotes the coproduct or disjoint union in $\Set$. This functor means that $\Tm$ takes one parameter $\Delta$ of sort $\Con$ and one parameter of sort $\Ty\;\Delta$, i.e. there is exactly one $\Tm$ for every pair of a $\Con$ called $\Delta$ in the model, and of a $\Ty$ associated to $\Delta$ in the model.
We build $\CC_3$ to be a pair of
\begin{itemize}
\item A model $(X_\Con,X_\Ty)$ from $\CC_2$
\item A family of sets $X_\Tm$ indexed by $H_\Tm(\Con)$
\end{itemize}
This category is isomorphic to the category composed of triples of
\begin{itemize}
\item A set $X_\Con$
\item A family of sets indexed by $X_\Con$ : $\left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con}$
\item A family of sets indexed by $(\Delta : X_\Con)$ and $X_\Ty(\Delta)$ :
$\left(\left(X_\Tm(\Delta,A)\right)_{A \in X_\Ty(\Delta)}\right)_{\Delta \in X_\Con}$
\end{itemize}
\paragraph{Category of models of the transformed GAT}
We have also seen in the introduction how was constructed the category of models of the transformed GAT. We will reconstruct it sort by sort.
\[
\boxed{\begin{array}{c}
\mathcal{O} : \Set \\
\El : \mathcal{O} \to \Set
\end{array}}
\]
We have seen in previous subsection that the category of models of this GAT is $\BB_0 = \TSet$. We will only add new \emph{constructors} to these models, and no new sorts.
\[
\boxed{\Con : \mathcal{O}}
\]
We are adding a constructor to the model. This constructor takes no parameters, which means we want to select one sort out of our model to be the $\Con$ sort. And we recall that the sorts of a model $X : \TSet$ are the elements of $X_\UU$.
Therefore a model of the above GAT with only one constructor is a model $X : \TSet$ along with an object $\Cstr^X_\Con \in X_\UU$. An object of $\BB_1$ is a pair of
\begin{itemize}
\item A functor $X : \TT \to \Set$
\item An element $\Cstr^X_\Con \in X_\UU$
\end{itemize}
A morphism $f$ from $(X,\Cstr^X_\Con)$ to $(X',\Cstr^{X'}_\Con)$ is simply a morphism of $\TSet$ from $X$ to $X'$ that respects the constructor, i.e. that is such that
\[
f_\UU(\Cstr^X_\Con) = \Cstr^{X'}_\Con
\]
\[
\boxed{\Ty : \underline{\Con} \to \mathcal{O}}
\]
Now the constructor we are adding takes one parameter. It means that for every element of $\El$ associated with $\Con$, there is an object in $\mathcal{O}$.
This constructor translates into an object
\[
\Cstr^X_\Ty : X_p^{-1}(\Cstr^X_\Con) \to X_\UU
\]
The codomain of this constructor means \enquote{Every object of $X_\El$ that are associated to the $\Con$ object of $X_\UU$ i.e. the object $\Cstr^X_\Con$}
This codomain is quite verbose, but we have another way of expressing it. In the final proof, we will have already constructed the functor $F_1 : \BB_1 \to \CC_1$, and we had a functor $H_\Ty : \CC_1 \to \Set$ that described the parameters of the sort declaration of $\Ty$ for the category $\CC_1$. Well, it turns out that $H_\Ty \circ F_1 : \BB_1 \to \Set$ describes exactly the codomain expressed above.
Therefore, the category $\BB_2$ consists of a pair of
\begin{itemize}
\item A model $(X,\Cstr^X_\Con)$ from $\BB_1$
\item A constructor $\Cstr^X_\Ty : H_iF_{i-1}(X,\Cstr^X_\Con) \to X_\UU$
\end{itemize}
One again, morphism $f$ from $(X,\Cstr^X_\Con,\Cstr^X_\Ty)$ to $(X',\Cstr^{X'}_\Con,\Cstr^{X'}_\Ty)$ are simply morphisms of $\BB_1$ from $(X,\Cstr^X_\Con)$ to $(X',\Cstr^{X'}_\Con)$ that respect the $\Cstr_\Ty$ constructor, i.e. that verifies, for every $\Gamma$ in $H_iF_{i-1}(X,\Cstr^X_\Con)$ that
\[
f_\UU(\Cstr^X_\Ty(\Gamma)) = \Cstr^{X'}_\Ty(f_\El(\Gamma))
\]
\[
\boxed{\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}}
\]
This one is made like the others, a model of $\BB_3$ is a pair of
\begin{itemize}
\item A model $(X,\Cstr^X_\Con,\Cstr^X_\Ty)$ of $\BB_2$
\item A constructor $\Cstr^X_\Tm$ that goes from $H_\Tm F_2 X$ to $X_\UU$
\end{itemize}
And morphism are restricted with the $\Tm$ constructor rule that states that for every $(\Delta,A)$ in $H_iF_{i-1}(X,\Cstr^X_\Con,\Cstr^X_\Ty)$
\[
f_\UU(\Cstr^X_\Tm(\Delta,A)) = \Cstr^{X'}_\Tm(f_\El(\Delta),f_\El(A))
\]
\paragraph{Constructing $G_3$}
We want to make functors connecting those two categories
\[
\begin{array}{l|l}
Y : \mathbf{\CC_3} & X : \mathbf{\BB_3} \\
& X : \TSet\\
Y_\Con : \Set & \Cstr^X_\Con : X_\UU \\
\left(Y_\Ty(\Gamma)\right)_{\Gamma \in Y_\Con} &
\Cstr^X_\Ty : H_\Ty F_{1} (X,\Cstr^X_\Con) \to X_\UU \\
\left(\left(Y_\Tm(\Delta,A)\right)_{A \in Y_\Ty(\Delta)}\right)_{\Delta \in Y_\Con} &
\Cstr^X_\Tm : H_\Tm F_{2} (X,\Cstr^X_\Con,\Cstr^X_\Ty) \to X_\UU \\
\end{array}
\]
Let's first look at the first component of the «result» $X = G_3(Y)$. It will be a functor of $\TSet$. We recall how $\TSet$ corresponds to the sort specification $(\mathcal{O},\El)$:
\begin{itemize}
\item $X_\UU$ is the set of all sorts
\item $X_\El$ is the set of all «objects» of those sorts
\item $X_p$ sends an object of a sort to the sort it corresponds to
\end{itemize}
Therefore, we will construct $X_\UU$ to be the disjoint union of all the indices of the sets inside $Y$, $X_\El$ will be the disjoint union of all the sets inside $Y$, and $X_p$ will keep track of what is the indice of a set.
\[\begin{array}{ccccccc}
X_\El & = &
Y_\Con & \oplus &
\displaystyle\coprod_{\Gamma \in Y_\Con}Y_\Ty(\Gamma) & \oplus &
\displaystyle\coprod_{\Delta \in Y_\Con}\coprod_{A \in Y_\Ty(\Delta)}Y_\Ty(\Delta,A) \\
\labeledupdownarrow{p}&&\labeledupdownarrow{}&&\labeledupdownarrow{\operatorname{proj}_1}&&\labeledupdownarrow{\operatorname{proj}_{2,1}}\\
X_\UU & = &
1 & \oplus &
Y_\Con & \oplus&
\displaystyle\coprod_{\Gamma \in Y_\Con}Y_\Ty(\Gamma)
\end{array}\]
where $\operatorname{proj}_1 : \coprod_{a\in A}B(a) \to A$ is the first projector of the coproduct, and $\operatorname{proj}_{2,1}$ get the index $ \coprod_{a \in A}B(a)$ of a double coproduct $\coprod_{a \in A}\coprod_{b\in B(a)} C(a,b)$
\vspace{.2ex}
Now, the constructors only have to give the right sort according to their arguments. So
\[\begin{array}{lcl}
\Cstr^X_\Con &=& \inj_1(\star)\\
\Cstr^X_\Ty(\Gamma) &=& \inj_2(\Gamma) \\
\Cstr^X_\Tm(\Delta,A) &=& \inj_3(\Delta,A)
\end{array}\]
where $\inj_i$ is the $i$-th injector of the direct sum $\oplus$.
\todo{Parler des morphismes ? Est-ce nécessaire ?}
\paragraph{Constructing $F_i$}
Now we will try to create an object $Y = F_i X$ of $\CC_3$ from an object $X$ of $\BB_3$.
\begin{itemize}
\item $Y_\Con$ should be the set of all objects of sort $\Con$. In the transformed GAT, it means all objects of $\El$ which have been created with the sort $\Cstr^X_\Con$. Therefore we have
\[
Y_\Con = X_p^{-1}(\left\{\Cstr^X_\Con\right\}) \subseteq X_\El
\]
\item $Y_\Ty(\Gamma)$ should be the set of all objects of sort $\Ty\;\Gamma$, which means all objects of $\El$ which have been created with the sort $\Cstr^X_\Ty(\Gamma)$. Therefore we have, for every $\Gamma$ in $Y_\Con$
\[
Y_\Ty(\Gamma) = X_p^{-1}(\{\Cstr^X_\Ty(\Gamma)\})
\]
This is well-defined, as
\[
\Gamma : Y_\Con = X(p)^{-1}(\{\Cstr^X_\Con\}) = H_\Ty(X(p)^{-1}(\{\Cstr^X_\Con\})) = H_\Ty F_1(X,\Cstr^X_\Con)
\]
\item In the same way, we define
\[
Y_\Tm(\Delta,A) = X_p^{-1}(\Cstr^X_\Tm(\Delta,A))
\]
which is also well-defined
\end{itemize}
\begin{remark}
We can see that the $F_3$ functor does not take into account elements of $X_\UU$ that are not \enquote{created} by one of the constructors, nor does it take into account the elements of $X_\El$ that are associated with those elements of $X_\UU$.
This will later justify that when one adds elements to the base object $X$ of a model of $\BB_i$, then the image model by $F_i$ is unchanged, i.e. $F_i \inj_\tl^i$ is an isomorphism.
\end{remark}
\paragraph{An adjunction}
I will not do the proof that $F_3 \vdash G_3$, but i will give moral reasons of why the adjunction holds.
An adjunction would be described by the following isomorphism:
\[
\Hom_{\BB_3}\left(G_3Y,X\right) \simeq \Hom_{\CC_3}\left(Y,F_3X\right)
\]
Recall that sorts of $G_3Y$ are all in the domain of some constructor, so therefore, a morphism of $\Hom_{\BB_3}(G_3Y,X)$ only acts on constructible sorts.
Also, the only sorts that remain in $F_3X$ are the one that were reached by some constructor of $X$, therefore a morphism of $\Hom_{\CC_3}(Y,F_3X)$ only acts on constructible sorts.
Therefore, both morphism are only descriptions of actions on constructible sorts into constructible sorts. That's how we create the isomorphism.
Morphisms of $\BB_3$ from $G_3Y$ to $X$ are morphisms of $\TSet$ that respect constructors. It means that
\paragraph{A coreflection}
If we look at $F_3G_3 Y$ for a model $Y$ of $\CC_3$, we remark that
\[\begin{array}{lcl}
F_3G_3(Y)_\Con &=& G(Y)_p^{-1}(\{\Cstr^{G(Y)}_\Con\})\\
&=& G(Y)_p^{-1}(\{\inj_1 \star\}) \\
&=& Y_\Con
\end{array}\]
and
\[\begin{array}{lcl}
F_3G_3(Y)_\Ty(\Gamma) &=& G(Y)_p^{-1}(\{\Cstr^{G(Y)}_\Ty(\Gamma)\})\\
&=& G(Y)_p^{-1}(\{\inj_2 \Gamma\}) \\
&=& proj_1^{-1}(\Gamma) \\
&=& \left\{(\Gamma',A) \in \coprod_{\Gamma' \in Y_\Con}Y_\Ty(\Gamma') \middle| \Gamma' = \Gamma\right\}\\
&\simeq& Y_\Ty(\Gamma)
\end{array}\]
and finally, with the same method, we get that
\[
F_3G_3(Y)_\Tm(\Delta,A) \simeq Y_\Tm(\Delta,A)
\]
And therefore $F_3G_3 \equiv \Id_{\CC_3}$.
\section{Constructing the coreflection}
Here is a figure that describes the recursive construction of some of the above objects
\begin{center}
% YADE DIAGRAM GlobalRecursiveConstruction.json
@ -189,6 +547,7 @@
We will often concatenate the two method above to create from a category $\mathcal{C}$ and a functor $H : \mathcal{C} \to \Set$ a new category $(X : \mathcal{C}) \times \left(\Set\middle/H(X)\right)$.
\paragraph{Slice category over a set}
\label{SetXSetXEquivalence}
When the category $\mathcal{C}$ is $\Set$, we have the equivalence
\[\Set/X \simeq \Set^X\]
@ -545,7 +904,6 @@
\]
It is a morphism of $\BB_i$ as it makes the following diagram commute:
\begin{center}
% YADE DIAGRAM TlInj1MorphismOfBi.json
% GENERATED LATEX
@ -1111,6 +1469,11 @@

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1
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9
Report/header.tex
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