Modified first part, constructing the category

Report/Bilibibio.bib

@@ -1,21 +1,12 @@
 @InProceedings{Fiore2008,
-	
 	author={Fiore, Marcelo},
-	
 	booktitle={2008 23rd Annual IEEE Symposium on Logic in Computer Science}, 
-	
 	title={Second-Order and Dependently-Sorted Abstract Syntax}, 
-	
 	year={2008},
-	
 	volume={},
-	
 	number={},
-	
 	pages={57-68},
-	
 	keywords={Algebra;Computer science;Mathematical model;Logic functions;Laboratories;MONOS devices;Sorting;abstract syntax;second-order syntax;dependently-sorted syntax;alpha-equivalence;variable binding;substitution;metavariable;meta-substitution;categorical algebra},
-	
 	doi={10.1109/LICS.2008.38}
 }
 
@@ -24,7 +15,7 @@
 	editor={"Baier, Christel and Dal Lago, Ugo"},
 	title={"Quotient Inductive-Inductive Types"},
 	booktitle={"Foundations of Software Science and Computation Structures"},
-	year={"2018"},
+	year = 2018,
 	publisher={"Springer International Publishing"},
 	address={"Cham"},
 	pages={"293--310"},
@@ -40,7 +31,7 @@
 	series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
 	ISBN =	{978-3-95977-285-3},
 	ISSN =	{1868-8969},
-	year =	{2023},
+	year =	2023,
 	volume =	{269},
 	editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
 	publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
@@ -55,10 +46,33 @@
 	journal = {Annals of Pure and Applied Logic},
 	volume = {32},
 	pages = {209-243},
-	year = {1986},
+	year = 1986,
 	issn = {0168-0072},
 	doi = {https://doi.org/10.1016/0168-0072(86)90053-9},
 	url = {https://www.sciencedirect.com/science/article/pii/0168007286900539},
 	author = {John Cartmell}
 }
 
+@phdthesis{SestiniPhD,
+	author  = {Filippo Sestini},
+	title   = {Bootstrapping Extensionality},
+	school  = {University of Nottingham},
+	year    = 2023,
+	month   = mar
+}
+
+@misc{AmbrusSzumiXie2sort,
+	author = {Ambrus Kaposi},
+	title = {Message to the Agda mailing list},
+	howpublished = {\url{https://lists.chalmers.se/pipermail/agda/2019/011176.html}},
+	year = 2019
+}
+
+@misc{nlab:reflective_subcategory,
+	author = {{nLab authors}},
+	title = {reflective subcategory},
+	howpublished = {\url{https://ncatlab.org/nlab/show/reflective+subcategory}},
+	note = {\href{https://ncatlab.org/nlab/revision/reflective+subcategory/116}{Revision 116}},
+	month = jul,
+	year = 2024
+}

Report/M2Report.tex

@@ -3,6 +3,10 @@
 
 \input{./header.tex}
 
+% po4a: environment remark
+% po4a: environment tikzpicture
+% po4a: environment property
+
 \title{Categorical semantics of the reduction of GATs to two-sorted GATs.
 	\\[1ex] \large Notes on my 4.5-month internship at the Laboratoire d'Informatique de l'École Polytechnique (Palaiseau, France)}
 \hypersetup{pdftitle={Categorical semantics of the reduction of GATs to two-sorted GATs}}
@@ -43,8 +47,8 @@
 	A model of this category is a triple
 	\begin{itemize}
 		\item A set $X_\Con : \Set$
-		\item A function $X_\Ty : X_\Con \to \Set$
+		\item A family of sets $\left(X_\Ty\left(\Gamma\right)\right)_{\Gamma \in _\Con}$
-		\item A function $X_\Tm : \displaystyle\coprod_{\Gamma : X_\Con} X_\Ty(\Gamma) \to \Set$
+		\item A family of sets $\left(X_\Tm\left(\Delta,A\right)\right)_{\Delta\in X_\Con,\: A \in X_\Ty\left(\Delta\right)}$
 	\end{itemize}
 	
 	\paragraph{Constructor specification}
@@ -72,13 +76,15 @@
 	
 	\paragraph{Two-sortification}
 	
-	There is a process that allows us to transform a GAT into a GAT with only two sorts. This process is applied for example by Philippo Sestini \inlinetodo{manque une citation}, who asserts that one can build back an initial model 
+	There is a process that allows us to transform a GAT into a GAT with only two sorts. This process is used by Philippo Sestini in his thesis \cite{SestiniPhD} refering the work of Zongpu Szumi Xie \cite{AmbrusSzumiXie2sort}:
 		
-	this transformation preserves the existence of the initial model.
+	\begin{quote}
+		Many instances of multi-sorted IITs [IITs are another type of GATs] can be reduced to equivalent two-sorted IITs, via a systematic reduction method originally observed by Zongpu (Szumi) Xie. We are not aware of a formal proof of this construction for arbitrary IITs, but we conjecture that it does apply to all instances of induction-induction and	consequently that it shows two-sorted IITs are enough to represent any specifiable	IIT.
+	\end{quote}
 	
-	The goal of this document is to prove semantically that this transformation creates an adjunction, and more precisely a reflective adjunction between the categories of models (and therefore preserving the existence of an initial model).
+	The goal of this document is to prove semantically that this transformation makes sense. More specifically, we prove that this transformation is a left adjunct functor of a coreflection. This is enough to prove what Sestini conjectured, i.e. that the initial object in the 2-sort category creates back the initial object of the primary category \cite[5. General]{nlab:reflective_subcategory}.
 	
-	We will now present this transformation. The sort specification of the GAT is always the same, and contains two sort declarations (as planned):
+	We will now present this transformation. The sort specification of the transformed GAT is always the same, and contains two sort declarations (as planned):
 	
 	\vspace{1em}
 	\begin{tabular}{p{0.37\textwidth}|p{0.5\textwidth}}
@@ -87,7 +93,7 @@
 	\end{tabular}
 	\vspace{1em}
 	
-	Category of models of this two-sort specification are intuitively the category of families of set $\FamSet$, composed of pairs $\left(X_0:\Set,X_1: X_0 \to \Set\right)$ (or $\coprod_{X : \Set}\Set$ in type theory).
+	Category of models of this two-sort specification are intuitively the category of families of set $\FamSet$, composed of pairs $\left(X_0:\Set,X_1: X_0 \to \Set\right)$.
 	
 	Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply underline to every parameter. For example, the Type Theory GAT presented above becomes that which follows:
 	
@@ -98,64 +104,16 @@
 		$\Tm : (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}$ &
 		For each object $\Gamma$ corresponding to the sort object $\Con$,
 		and for every object $A$ corresponding to the sort object $\Ty\;\Gamma$, another sort object called \enquote{$\Tm\;\Gamma\;A$}\\
-		$\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ &
+		$\operatorname{unit} : (\Gamma : \underline{\Con}) \to \underline{\Ty\;\Gamma}$ &
 		For each object $\Gamma$ corresponding to the sort object $\Con$, an object called \enquote{$\operatorname{unit} \Gamma$} corresponding to the sort object $\Ty\;\Gamma$\\
-		$\operatorname{eq}: (\Gamma : \Con) \to (A : \Ty\;\Gamma) \to$ \newline
+		$\operatorname{eq}: (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Gamma}) \to$ \newline
-		$\qquad\Tm\;\Gamma A \to \Tm\;\Gamma A \to \Ty\;\Gamma$ &
+		$\qquad\underline{\Tm\;\Gamma A} \to \underline{\Tm\;\Gamma A} \to \underline{\Ty\;\Gamma}$ &
 		$\dots$
 	\end{tabular}
 	
-	\paragraph{Fiore's categories}
+	\paragraph{$\FamSet$ as functors}
-	Fiore \cite{Fiore2008} describes \emph{sort specifications} as countable simple direct categories (i.e. countable categories where all the arrows follow an unique direction and hom-sets are finite). The models of a GAT then are the presheaves over that category $S$: $\left[S,\Set\right]$.
-	
-	One can understand the correspondance between those categories and sort specifications as follows:
-	\begin{itemize}
-		\item An object of the category is a sort of the specification.
-		\item An arrow $x$ from an object $s$ to an object $s'$ is a parameter of the sort declaration of $s$ of the for $(x : s' \dots)$.
-		\item The parameter $y$ of a parameter $x$ of a sort specification (i.e. the sort declaration parameter has the form $(x: s' \dots \left[y=z\right] \dots)$) is given by $z = x \circ y$.
-	\end{itemize}
-	
-	\begin{remark}
-		We ignore in this definition identity arrows, and we will do so in the rest of this document. Identities are the only arrows that are not «directed» in the direct category.
-		
-		Interpreting the identity arrow would mean having a parameter of type $s$ to construct the sort $s$. which loops in a self-dependency.
-		
-		You can assume in the rest of the document that the formalizations \enquote{all arrows} or \enquote{the arrows} pointing to/from exclude identity arrows.
-	\end{remark}
-	
-	\todo{Éventuellement changer tous les paramètres par la forme complète, exemple 
-	\[
-	\operatorname{eq}: (\Gamma : \Con) \to (A : \Ty \left[\Gamma=\Gamma\right]) \to \Tm \left[\Gamma=\Gamma\right] \left[A=A\right] \to \Tm \left[\Gamma=\Gamma\right] \left[A=A\right] \to \Ty \left[\Gamma=\Gamma\right]
-	\]
-	C'est bien plus verbeux et en pratique pas utilisé, mais permet de mieux voir la «composition» dans la catégorie de Fiore.}
-	\todo{Est-ce qu'on fait une notation \enquote{arrow*} pour dire «flèche qui n'est pas l'identité» pour plus de rigueur ?}
-	
-	For example the category version of the specification of sorts of Type Theory given above is defined as:
-	
-	\begin{itemize}
-		\item There is three objects called $\Con$,$\Ty$, and $\Tm$.
-		\item The arrows are defined as
-		\begin{itemize}
-			\item There is no arrow going out of $\Con$
-			\item There is one arrow going out of $\Ty$: $\Gamma$ pointing to $\Con$.
-			\item There is two arrows going out of $\Tm$: $\Delta$ pointing to $\Con$ and $\Gamma$ pointing to $\Ty$.
-		\end{itemize} 
-		\item The $\Gamma$ parameter of $\Ty$ in the parameter $A$ of $\Tm$ is $\Delta$. Therefore, we have $\Delta = A \circ \Gamma$.
-	\end{itemize}
-	
-	The category is pictured below:
-	
-	\begin{center}
-		% YADE DIAGRAM B1.json
-		% GENERATED LATEX
-		\input{graphs/B1.tex}
-		% END OF GENERATED LATEX
-	\end{center}
-	
-	\paragraph{Fiore's category of the 2-sorts specification}
-	
-	If compute the small category associated to the two-sort specification described above, we obtain the simple category with two objects and one arrow between them. We call this category $\TT$ and write the objects as follows:
 	
+	In the rest of the document, we will denote the simple category containing two elements and one non-identity arrow between them as $\TT$. The objects and arrow of this category are pictured below.
 	
 	\begin{center}
 		% YADE DIAGRAM G0.json
@@ -164,11 +122,21 @@
 		% END OF GENERATED LATEX
 	\end{center}
 	
-	The category of presheaves over this category is equivalent to the category $\FamSet$.
+	The functors over this categories are equivalent to families of sets, using the following mapping :
+	
+	\[
+	\begin{array}{l|l}
+		X_\UU = X_0 & X_0 = X_\UU \\
+		X_\El = \displaystyle\coprod_{A\in X_0}X_1(A) & X_1 = A \mapsto X_p^{-1}(\{A\})\\
+		X_p = (A,B) \mapsto A &
+	\end{array}
+	\]
+	
+	Therefore the categories of sorts of the transformed GATs will be built atop of the category $\TSet$ rather than atop of the category $\FamSet$ as it makes the formal proofs more elegant.
 	
 	\paragraph{Goal}
 	
-	The goal of this document is to make a relation between the category of models of the GAT $\left[S,\Set\right]$ and the category of models of the two-sortified GAT $\BB$. This relation will be an adjunction $F \vdash G$ that we will prove to be a coreflection.
+	The goal of this document is to make a relation between the category of models of the GAT $\CC$ and the category of models of the two-sortified GAT $\BB$. This relation will be an adjunction $F \vdash G$ that we will prove to be a coreflection.
 	
 	The category $\BB$ is built with an adjunction $R \vdash L$ to the category of models of the simple two-sort specification of sorts $\TSet$.
 	
@@ -181,100 +149,93 @@
 	
 	\subsection{Constructing the categories}
 	
-	We will construct both categories $S$ and $\BB$ recursively, adding new sorts one by one.
+		We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one.
-	The categories $S_i$ 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 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 first step of our recursion is the trivial adjunction $\lambda . \star \vdash \lambda . 1$ between the categories $\TSet = \BB_0$ and $\left[S_0,\Set\right] = \left[\emptyset,\Set\right] = 1$.
+		The overall construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
 		
-	Then we construct the category $S_i$ as a full supercategory of $S_{i-1}$ (and so we have a ff injection functor $I_i : S_{i-1} \to S_i$).
+		\begin{center}
+			% YADE DIAGRAM G1.json
+			% GENERATED LATEX
+			\input{graphs/G1.tex}
+			% END OF GENERATED LATEX
+		\end{center}
 		
-	We construct at the same time the category $\BB_i$ along with an adjunction $R_{i-1}^i \vdash L_{i-1}^i$ with $\BB_{i-1}$.
+		The first step of our recursion is the trivial adjunction $\lambda . \star \vdash \lambda . 1$ between the categories $\BB_0 = \TSet$ and $\CC_0 = 1$.
 		
-	The construction is summarized in the following diagram:
+		\subsubsection{Constructing $\CC_i$}
 		
-	\begin{center}
+		We construct the category $\CC_i$ as the following pair:
-		% YADE DIAGRAM G1.json
+		\[
-		% GENERATED LATEX
+			\CC_i = (X : \CC_{i-1}) \times \Set^{\Hom(O_i,X)} \qquad\text{(this is a dependent coproduct)}
-		\input{graphs/G1.tex}
+		\]
-		% END OF GENERATED LATEX
+		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.
-	\end{center}
+		
+		For example, for our type theory example, we first have
+		\[
+			O_1 = \star \in \operatorname{Obj}(\CC_0) = \operatorname{Obj}(1)
+		\]
+		so $\Hom(O_1,X) = 1$, which corresponds to the fact that $\Con$ takes no parameter.
+		
+		Therefore $\CC_1 = 1 \times \Set^1 = \Set$
+		
+		Then, we take the singleton object $O_2 = 1$ (this means, that types need \emph{one} context to be built), and so, for a set $X_\Con$, $\Hom(O_2,X_\Con) \cong X_\Con$, which corresponds to the fact that $\Ty$ take one $\Con$ as a parameter.
+		
+		Therefore $\CC_2 = (X:\Set) \times \Set^X \cong \FamSet$.
+		
+		Finally, we take the object $O_3 = (1, \lambda \star . 1)$ (this means that terms need \emph{one} context, and \emph{one} type of that context). With this object, for a pair $(X_\Con,X_\Ty)$ in $\CC_2$, we have $\Hom(O_3,(X_\Con,X_\Ty)) \cong \left(\Gamma: X_\Con, A: X_\Ty(\Gamma)\right)$.
+		
+		The final category $\CC_3$ is composed of triples $(X_\Con: \Set, X_\Ty : X_\Con \to \Set, X_\Tm : (\Delta: X_\Con) \to X_\Ty(\Delta) \to \Set)$
+		
+		\begin{remark}
+			There is a way of getting the object $O_i$ from the syntax, which is given in \autoref{sec:CtoSSetFiore}.
+		\end{remark}
+		
+		\subsubsection{Constructing $\BB_i$}
+		
+		\paragraph{The category} We construct the category $\BB_i$ as follows.
+		
+		An object of $\BB_i$ is
+		\begin{itemize}
+			\item an object $X$ of $\BB_{i-1}$
+			\item a \enquote{sort constructor} $\Cstr_i$ as a function $\Hom_{\BB_{i-1}} (G_{i-1}O_i,X) \to (R_0^{i-1}X)_\UU$
+			\newline
+			where $O_i$ is the object of $\CC_{i-1}$ that describe the sort constructor being processed, and $G_{i-1}$ is the left part of the adjunction $\CC_{i-1} \to \BB_{i-1}$ that we are defining recursively at the same time.
+		\end{itemize}
 		
 		
-	\subsubsection{Fiore's category}
+		A morphism $(X,\Cstr_i) \to (X',\Cstr'_i)$ of $\BB_i$ is a morphism $f : X \to X'$ in $\BB_{i-1}$ such that the following diagram commutes.
 		
-	In order to construct the $i$-th sort, we use a finite functor $\Gamma_i : S_{i-1} \to \Set$ describing entirely the sort declaration.
+		\begin{center}
-	This functor is to be understood as $\Gamma_i(a)$ is the set of parameters of type $a$ for our new sort. In the above example, we would have $\Gamma_\Ty(\Con) = \{"\Gamma"\} = 1$ and $\Gamma_\Tm(\Con) = \{\Delta\}$,$\Gamma_\Tm(\Ty) = \{"A"\}$,$\Gamma_\Tm(\Gamma) = \left["A" \mapsto "\Delta"\right]$.
+			% YADE DIAGRAM D1.json
+			% GENERATED LATEX
+			\input{graphs/D1.tex}
+			% END OF GENERATED LATEX
+		\end{center}
 		
-	Then, to construct $S_i$, we add one object $i$ to $S_{i-1}$, along with morphisms $x : i \to a$ for every $x \in \Gamma_i(a)$ for every $a$ in $S_{i-1}$. We also add equalities 
+		Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
-	$s \circ x = x'$ for every $s : b \to a$ and $x \in \Gamma_i(a)$ and $x' \in \Gamma_i(b)$ where $\Gamma_i(s)(x') = x$.
 		
-	\begin{remark}
+		\paragraph{The adjunction}
-		We have that $\Hom_{S_i}(a,b) = \Gamma_b(a)$ or $(a/S_i)* \equiv \Gamma_a$.\inlinetodo{C'est sûr la deuxième partie ?}
+		We also define a functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$ that sends objects and morphisms to their first component. This functor is a \emph{right adjunct} of another functor we call $L_{i-1}^i$.
 		
-		This equality allows us to construct the $\Gamma_i$ functors from the final $S$ category.
+		As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will create the two following syntactic sugars for the composed adjunctions.
-	\end{remark}
+		\[\begin{array}{c}
-	
+		R_i^j = R_{i}^{i+1} \circ R_{i+1}^j = R_{i}^{j-1} \circ R_{j-1}^{j} = R_{i}^{i+1} \circ ... \circ R_{j-1}^{j}\\
-	\subsubsection{2sort category}
+		L_i^j = L_{j-1}^{j} \circ L_{i}^{j-1} = L_{i+1}^{j} \circ L_{i}^{i+1} = L_{j-1}^{j} \circ ... \circ L_{i}^{i+1}
-	
+		\end{array}\]
-	To start our series of categories, we use the category of models of the two-sort specification of sorts $\BB_0 := \TSet$.
-	
-	Then, we recursively add constructors, constructing categories $\BB_1$,$\BB_2$, etc.
-	
-	For the $i$-th constructor, we define the objects of $\BB_i$ as pairs of
-	
-	\begin{itemize}
-		\item an object $X$ of $\BB_{i-1}$
-		\item a \enquote{sort constructor} $\Cstr_i$ as a function $\Hom_{\BB_{i-1}} (G_{i-1}\Gamma_i,X) \to (R_0^{i-1}X)_\UU$
-		\newline
-		where $\Gamma_i$ is the functor $S_{i-1} \to \Set$ that describe the sort constructor being processed, and $G_{i-1}$ is the left part of the adjunction $\left[S_{i-1}, \Set\right] \to \BB_{i-1}$ that we are defining recursively at the same time.
-	\end{itemize}
-	
-	
-	Morphisms $(X,\Cstr_i) \to (X',\Cstr'_i)$ in $\BB_i$ are morphisms $f : X \to X'$ in $\BB_{i-1}$ such that the following diagram commutes.
-	
-	\begin{center}
-		% YADE DIAGRAM D1.json
-		% GENERATED LATEX
-		\input{graphs/D1.tex}
-		% END OF GENERATED LATEX
-	\end{center}
-	
-	Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
-seeing
-	We also define a functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$ that sends objects and morphisms to their first component. This functor is a \emph{right adjunct} of another functor we call $L_{i-1}^i$.
-	
-	As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will create the two following syntactic sugars for the composed adjunctions.
-	\[
-	R_i^j = R_{i}^{i+1} \circ R_{i+1}^j = R_{i}^{j-1} \circ R_{j-1}^{j} = R_{i}^{i+1} \circ ... \circ R_{j-1}^{j}
-	\]
-	\[
-	L_i^j = L_{j-1}^{j} \circ L_{i}^{j-1} = L_{i+1}^{j} \circ L_{i}^{i+1} = L_{j-1}^{j} \circ ... \circ L_{i}^{i+1}
-	\]
-	
-	We will also denote $\eta_i^j : \mathbf{1} \to R_i^j L_i^j$ and $\varepsilon_i^j : L_i^j R_i^j \to \mathbf{1}$ to be the unit and counit of the adjunction $R_i^j \vdash L_i^j$.
-	
-	
-	
-	We know that this category has a coproduct, that we will denote $\oplus_i$ or just $\oplus$ when there is no ambiguity. We will also denote as $\inj_1^i : X \to X \oplus Y$ (resp. $\inj_2^i : Y \to X \oplus Y$) the first (resp. second) injector of the coproduct of $\BB_i$. For every morphisms $f : X \to Z$ and $g : Y \to Z$, we will denote with $\{f;g\}$ the unique morphism from $X \oplus Y$ to $Z$ such that $\{f;g\} \circ \inj^i_1 = f$ and $\{f;g\} \circ \inj^i_2 = g$.
-
-	\begin{remark}
-		This adjunction and the existence of a coproduct comes from seeing $\BB_i$ being a category of algebras in $\BB_{i-1}$ over the morphism $inj_1 : G_{i-1}\Gamma_i \to G_{i-1}\Gamma_i \oplus L_0^{i-1} y\UU$.
-	\end{remark}
-	
-	\subsubsection{Summary}
-	
-	Here is a graph summarizing the categories and functors.
-	We have constructed two chains of categories $\BB_0$,$\BB_1$,... and $S_0$,$S_1$,...
-	
-	The categories $\BB_{i-1}$ and $\BB_{i}$ are in an adjunction written $R_{i-1}^i \vdash L_{i-1}^i$.
-	
-	We will give in the next part the construction of the adjunction $F_i \vdash G_i$ at the step $i$. The functor $G_{i-1}$ is used in the definition of $\BB_i$, so the two recurrences have to be done at the same time.
 		
+		We will also denote $\eta_i^j : \mathbf{1} \to R_i^j L_i^j$ and $\varepsilon_i^j : L_i^j R_i^j \to \mathbf{1}$ to be the unit and counit of the adjunction $R_i^j \vdash L_i^j$.
 		
+		\paragraph{The coproduct}
+		For an object $X$ in $\BB_i$ and $Y$ in $\BB_0$, there is a coproduct $X \oplus_i L_0^i Y$ in the category $\BB_{i-1}$. We will denote as $\inj_1^i : X \to X \oplus L_0^iY$ (resp. $\inj_2^i : L_0^iY \to X \oplus L_0^iY$) the first (resp. second) injector of the coproduct of $\BB_i$. For every morphism $f : X \to Z$ and $g : L_0^iY \to Z$, we will denote with $\{f;g\}$ the unique morphism from $X \oplus L_0^iY$ to $Z$ such that $\{f;g\} \circ \inj^i_1 = f$ and $\{f;g\} \circ \inj^i_2 = g$.
 	
+		\begin{remark}
+			This adjunction and the existence of a coproduct comes from seeing $\BB_i$ being a category of algebras in $\BB_{i-1}$ over the morphism $inj_1 : G_{i-1}O_i \to G_{i-1}O_i \oplus L_0^{i-1} y\UU$.
+		\end{remark}
 	
 	
 	\subsection{Constructing the adjunction}
+		We will now construct the adjunction $F_i \vdash G_i$ at the step $i$.
 	
 	\subsubsection{Hypotheses}
 	
@@ -508,7 +469,58 @@ seeing
 		
 	\subsection{Proof of H3}
 	
-		\subsection{Infinite construction of $\BB_i$}
+	
+	\section{Misc}
+	
+	\subsection{Fiore's Category - Fibration of the category of sorts}
+	
+	Fiore \cite{Fiore2008} describes \emph{sort specifications} as countable simple direct categories (i.e. countable categories where all the arrows follow an unique direction and hom-sets are finite). The models of a GAT then are the presheaves over that category $S$: $\left[S,\Set\right]$.
+	
+	One can understand the correspondance between those categories and sort specifications as follows:
+	\begin{itemize}
+		\item An object of the category is a sort of the specification.
+		\item An arrow $x$ from an object $s$ to an object $s'$ is a parameter of the sort declaration of $s$ of the for $(x : s' \dots)$.
+		\item The parameter $y$ of a parameter $x$ of a sort specification (i.e. the sort declaration parameter has the form $(x: s' \dots \left[y=z\right] \dots)$) is given by $z = x \circ y$.
+	\end{itemize}
+	
+	\begin{remark}
+		We ignore in this definition identity arrows, and we will do so in the rest of this document. Identities are the only arrows that are not «directed» in the direct category.
+		
+		Interpreting the identity arrow would mean having a parameter of type $s$ to construct the sort $s$. which loops in a self-dependency.
+		
+		You can assume in the rest of the document that the formalizations \enquote{all arrows} or \enquote{the arrows} pointing to/from exclude identity arrows.
+	\end{remark}
+	
+	\todo{Éventuellement changer tous les paramètres par la forme complète, exemple 
+		\[
+		\operatorname{eq}: (\Gamma : \Con) \to (A : \Ty \left[\Gamma=\Gamma\right]) \to \Tm \left[\Gamma=\Gamma\right] \left[A=A\right] \to \Tm \left[\Gamma=\Gamma\right] \left[A=A\right] \to \Ty \left[\Gamma=\Gamma\right]
+		\]
+		C'est bien plus verbeux et en pratique pas utilisé, mais permet de mieux voir la «composition» dans la catégorie de Fiore.}
+	\todo{Est-ce qu'on fait une notation \enquote{arrow*} pour dire «flèche qui n'est pas l'identité» pour plus de rigueur ?}
+	
+	For example the category version of the specification of sorts of Type Theory given above is defined as:
+	
+	\begin{itemize}
+		\item There is three objects called $\Con$,$\Ty$, and $\Tm$.
+		\item The arrows are defined as
+		\begin{itemize}
+			\item There is no arrow going out of $\Con$
+			\item There is one arrow going out of $\Ty$: $\Gamma$ pointing to $\Con$.
+			\item There is two arrows going out of $\Tm$: $\Delta$ pointing to $\Con$ and $\Gamma$ pointing to $\Ty$.
+		\end{itemize} 
+		\item The $\Gamma$ parameter of $\Ty$ in the parameter $A$ of $\Tm$ is $\Delta$. Therefore, we have $\Delta = A \circ \Gamma$.
+	\end{itemize}
+	
+	The category is pictured below:
+	
+	\begin{center}
+		% YADE DIAGRAM B1.json
+		% GENERATED LATEX
+		\input{graphs/B1.tex}
+		% END OF GENERATED LATEX
+	\end{center}
+	
+	\subsection{Infinite construction of $\BB_i$}
 	\[
 	\BB_i := \left(X : \TSet, \Cstr : (a : S_{i-1}) \to \Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,R_{a-1}^i(\this)) \to X(\UU)\right)
 	\]
@@ -569,6 +581,8 @@ seeing
 	
 	\subsection{Overview}
 	
+	\subsubsection{$\CC$ as presheaf category}
+	\label{sec:CtoSSetFiore}
 	We use the specification of sorts definition of Fiore \cite{Fiore2008}.
 	
 	A specification of sorts is given by a sequence of functors $\Gamma_i : S_{i-1} \to \Set$. We construct the category $S_{i+1}$ by adding a single object $\alpha_{i+1}$ to the category $S_{i}$, along with morphisms $f : \alpha_j \to \alpha_{i+1}$ for $f \in \Gamma_{i+1}(\alpha_j)$ and $j \leq i$. The morphisms follow the composition condition, describing that every pair of morphisms $f : \alpha_j \to \alpha_{i+1}$ and $g : \alpha_k \to \alpha_{i+1}$ (i.e. $f\in\Gamma_{i+1}(\alpha_k)$ and $g\in\Gamma_{i+1}(\alpha_j)$) and for every morphism of $S_{i}$ $h : \alpha_j \to \alpha_k$, we have $\Gamma_{i+1}(h)(f) \circ f = g$.
@@ -583,6 +597,20 @@ seeing
 	
 	So we can construct the base category, which is that of families of sets.
 	
+	
+	In order to construct the $i$-th sort, we use a finite functor $\Gamma_i : S_{i-1} \to \Set$ describing entirely the sort declaration.
+	
+	This functor is to be understood as $\Gamma_i(a)$ is the set of parameters of type $a$ for our new sort. In the above example, we would have $\Gamma_\Ty(\Con) = \{"\Gamma"\} = 1$ and $\Gamma_\Tm(\Con) = \{\Delta\}$,$\Gamma_\Tm(\Ty) = \{"A"\}$,$\Gamma_\Tm(\Gamma) = \left["A" \mapsto "\Delta"\right]$.
+	
+	Then, to construct $S_i$, we add one object $i$ to $S_{i-1}$, along with morphisms $x : i \to a$ for every $x \in \Gamma_i(a)$ for every $a$ in $S_{i-1}$. We also add equalities 
+	$s \circ x = x'$ for every $s : b \to a$ and $x \in \Gamma_i(a)$ and $x' \in \Gamma_i(b)$ where $\Gamma_i(s)(x') = x$.
+	
+	\begin{remark}
+		We have that $\Hom_{S_i}(a,b) = \Gamma_b(a)$ or $(a/S_i)* \equiv \Gamma_a$.\inlinetodo{C'est sûr la deuxième partie ?}
+		
+		This equality allows us to construct the $\Gamma_i$ functors from the final $S$ category.
+	\end{remark}
+	
 	\section{Summary}
 		
 		\lipsum[2-3]
@@ -704,3 +732,7 @@ seeing
 	
 	
 \end{document} 
+
+
+
+

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Report/header.tex

@@ -102,6 +102,7 @@
 \newcommand\BB{{\ensuremath{\mathcal{B}}}}
 \newcommand\TT{{\ensuremath{\mathcal{T}}}}
 \newcommand\UU{{\ensuremath{\mathcal{U}}}}
+\newcommand\CC{{\ensuremath{\mathcal{C}}}}
 \newcommand\El{{\ensuremath{\operatorname{\mathcal{E}l}}}}
 \newcommand\ii{{\ensuremath{\mathbf{i}}}}
 \newcommand\Con{{\ensuremath{\operatorname{Con}}}}

Report/yade-preamble.tex

@@ -2,6 +2,7 @@
 \newcommand\BB{{\ensuremath{\mathcal{B}}}}
 \newcommand\TT{{\ensuremath{\mathcal{T}}}}
 \newcommand\UU{{\ensuremath{\mathcal{U}}}}
+\newcommand\CC{{\ensuremath{\mathcal{C}}}}
 \newcommand\El{{\ensuremath{\operatorname{\mathcal{E}l}}}}
 \newcommand\ii{{\ensuremath{\mathbf{i}}}}
 \newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}