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Report/M2Report.tex
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@ -38,53 +38,43 @@
\vspace{1em}
\renewcommand\arraystretch{1.5}
\begin{tabular}{l|p{.5\textwidth}}
$\Con : \Set$ & A set of contexts\\
$\Ty : (\Gamma : \Con) \to \Set$ & For each context $\Gamma$, a set of types in this context\\
$\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set$ & For each context $\Gamma$ and each type $A$ in this context, a set of terms of this type.
$\Con : \Set$ & A set of contexts $X_\Con$\\
$\Ty : (\Gamma : \Con) \to \Set$ & For each context $\Gamma \in X_\Con$, a set $X_\Ty(\Gamma)$ of types in this context\\
$\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set$ & For each context $\Gamma \in X_\Con$ and each type $A \in X_\Ty(\Gamma)$ in this context, a set $X_\Tm(\Gamma,A)$ of terms of this type.
\end{tabular}
\vspace{1em}
A model of this category is a triple
\begin{itemize}
\item A set $X_\Con$ of contexts
\item A family of sets $\left(X_\Ty\left(\Gamma\right)\right)_{\Gamma \in _\Con}$ of types, indexed by contexts
\item A family of sets $\left(X_\Tm\left(\Delta,A\right)\right)_{\Delta\in X_\Con,\: A \in X_\Ty\left(\Delta\right)}$ of terms, indexed by contexts and types.
\end{itemize}
A sort declaration therefore describes the sets that the model contains.
\paragraph{Constructor specification}
We can also add constructors to a sort specification to make a complete GAT.
We can also add constructors to a sort specification to make a complete GAT. Those constructors rather describe elements of the sets contained in the model, previously defined by the sort declaration.
For example, for the previous sort specification, one can add the following constructors:
\vspace{1em}
\renewcommand\arraystretch{1.5}
\begin{tabular}{p{.37\textwidth}|p{.6\textwidth}}
$\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ & In any context $\Gamma$, a type of $\Ty\;\Gamma$ called unit.\\
$\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ & In any context $\Gamma \in X_\Con$, an element$\operatorname{unit}\;\Gamma \in X_\Ty(\Gamma)$.\\
$\operatorname{tt}: (\Gamma : \Con) \to \Tm\;\Gamma\;(\operatorname{unit}\;\Gamma)$ & In any context $\Gamma$, we have a term whose type is the $\operatorname{unit}$ of this context ($\operatorname{unit}\;\Gamma$).
$\operatorname{tt}: (\Gamma : \Con) \to \Tm\;\Gamma\;(\operatorname{unit}\;\Gamma)$ & In any context $\Gamma \in X_\Con$, an element $\operatorname{tt}\;\Gamma \in X_\Tm(\Gamma,\operatorname{unit}\;\Gamma)$.
\end{tabular}
\vspace{1em}
This adds to the previous model two functions that \enquote{points} one element of the sets
\begin{itemize}
\item For each $\Gamma \in X_\Con$, an element $\operatorname{unit}\;\Gamma \in X_\Ty(\Gamma)$
\item For each $\Gamma \in X_\Con$, an element $\operatorname{tt}\;\Gamma \in X_\Tm(\Gamma,\operatorname{unit}\;\Gamma)$
\end{itemize}
Sort declarations describe the sets that the model contains, whereas the constructors describe elements of these sets.
\paragraph{Two-sortification}
There is a process that allows us to transform a GAT into a GAT with only two sorts.
It was observed\cite{AmbrusSzumiXie2sort} that one can transform any GAT into a GAT with only two sorts. We will present this transformation.
The sort specification of the transformed GAT is always the same, and contains these two sort declarations:
\vspace{1em}
\begin{tabular}{p{0.37\textwidth}|p{0.5\textwidth}}
$\UU : \Set$ & The set of sorts \\
$\El : \mathcal{O} \to \Set$ & For every sort object $o$ in the set of sorts, a set called $\El(o)$ of objects corresponding to the sort object. We will write this set $\underline{o}$ rather than $\El(o)$.
$\El : \mathcal{O} \to \Set$ & For every sort object $o$ in the set of sorts, a set called $\El(o)$ of objects corresponding to the sort object.
\end{tabular}
\vspace{1em}
Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply $\El$ to every parameter. For example, the Type Theory GAT presented above becomes that which follows:
Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply $\El$ to every parameter. We write $\underline{o}$ rather than $\El(o)$ in order to ease reading. For example, the Type Theory GAT presented above becomes that which follows:
\begin{tabular}{p{0.4\textwidth}|p{0.5\textwidth}}
$\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\
@ -92,28 +82,30 @@
For each object $\Gamma$ corresponding to the sort object $\Con$, another sort object called \enquote{$\Ty\;\Gamma$} \\
$\Tm : (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}$ &
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 "$\Tm\;\Gamma\;A$"\\
and for every object $A$ corresponding to the sort object $\Ty\;\Gamma$, another sort object called \enquote{$\Tm\;\Gamma\;A$}\\
$\operatorname{unit} : (\Gamma : \underline{\Con}) \to \underline{\Ty\;\Gamma}$ &
For each object $\Gamma$ corresponding to the sort object $\Con$, an object called "$\operatorname{unit} \Gamma$" corresponding to the sort object $\Ty\;\Gamma$\\
$\operatorname{tt}: (\Gamma : \underline{\Con}) \to \underline{\Tm\;(\operatorname{unit}\;\Gamma)}$ &
$\dots$
\end{tabular}
This process has been observed by Zongpu Szumi Xie \cite{AmbrusSzumiXie2sort}, and Philippo Sestini used it in his thesis \cite{SestiniPhD}:
This process has not been formally justified, although it has been used like in Phipippo Sestini's thesis.
\todo{choose}
\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.
Many instances of multi-sorted IITs [IITs are variants 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}
\paragraph{Goal}
The goal my internship was to formally study this transformation, and to try to find a relation between the semantics of a GAT and its two-sorted version.
The goal of my internship was to study and undestrand the relationship between the categories of models of an original GAT and the category of models of the transformed GAT, in order to legitimate this transformation.
We managed to construct a coreflection between a category of models and the category of models of the transformed GAT.
The existence of this coreflection is enough to prove what Sestini conjectured.
We constructed a coreflection between those two categories, whose formal definition is given in next section. It consists of an adjunction $F \vdash G$ between the category $\CC$ of the models of the GAT and the category $\BB$ of the models of the two-sortified GAT, where G is full and faithful.
We will give a formal definition of this coreflection in next section. It will consist of an adjunction $F \vdash G$ between the category $\CC$ of the models of the GAT and the category $\BB$ of the models of the two-sortified GAT, where G is full and faithful.
The category $\BB$ will be built with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$. This forgetful $R$ functor is a composition of monadic functors, one for each sort constructor.
The category $\BB$ is equipped with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$. This forgetful $R$ functor is a composition of monadic functors, one for each sort declaration.
\todo{J'ai toujours pas regardé les terms constructors, je peux le mettre, mais je ne serais pas sûr de pouvoir le défendre}
\begin{center}
% YADE DIAGRAM G1-0.json
@ -128,7 +120,7 @@
\paragraph{Category of models of the two-sort sort specification}
The usual way of defining the category of models of the two-sort specification $\BB_0$ is by taking the category of families of sets. However, in order to have more elegant constructions, we will use a the category of models of the two-sort specification the category $\TSet$ of presheaves over the category with one arrow.
The usual way of defining the category of models of the two-sort specification $\BB_0$ is by taking the category of families of sets. However, we will rather use a the category of models of the two-sort specification the category $\TSet$ of presheaves over the category with one arrow.
In the rest of the document, we will denote this category with one arrow as $\TT$. The objects and arrow of this category are pictured below.
@ -156,7 +148,7 @@
The morphism $(X,Y) \to (X',Y')$ is therefore a pair of a morphism $f : X \to X'$ in $\mathcal{C}$ and a morphism $g : F(f)(Y) \to Y'$ in $H(X')$.
We will denote this category $(X : \mathcal{C}) \times F(X)$ as its objects are pairs. It can some times be found written as $\int^\mathcal{C} F$ or $\sqint^\mathcal{C} F$
We will denote this category $(X : \mathcal{C}) \times F(X)$ as its objects are pairs.
\paragraph{Slice category}
For a category $\mathcal{C}$ and $X$ an object in that category, the slice category (or over category) $\mathcal{C}/A$ is a category whose objects are pairs of
@ -169,9 +161,9 @@
We can deduce a functor $\left(\mathcal{C}/\dash\right) : \mathcal{C} \to \Cat$ from this construction.
If the category $\mathcal{C}$ is $\Set$, we have that $\Set/X \cong \Set^X$.
If the category $\mathcal{C}$ is $\Set$, we have the equivalence $\Set/X \simeq \Set^X$.
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 $(X : \mathcal{C}) \times \left(\Set\middle/H(X)\right)$.
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{$\Hbar$ functor}
@ -182,14 +174,14 @@
The morphisms are translated as-is.
\begin{remark}
This functor can be constructed using the property of the Grothendieck construction
This functor can be constructed using the formal construction of the Grothendieck construction as a pullback in the category of categories $\Cat$
\end{remark}
\subsection{Constructing the categories}
We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one for each constructor. At each recursion step, we will build the categories, the adjunction, and keep some invariants that will be stated in \autoref{sec:hypotheses}.
We will construct both categories $\CC$ and $\BB$ sort declaration by sort declaration in one big recursive process. At each step, we will build the categories $\CC_i$ and $\BB_i$, the adjunction $F_i \vdash G_i$, and keep some invariants that will be stated in \autoref{sec:hypotheses}.
At the i-th recursion step, we will build the category $\CC_i$ which is the category of models of the $i$ first sorts of the sort specification. $\BB_i$ will samewise be the category of models of the 2-sorted $i$ first sorts of the sort specification.
At the i-th recursion step, we will build the category $\CC_i$ which is the category of models of the $i$ first sorts of the sort specification. Likewise, $\BB_i$ will be the category of models of the 2-sorted $i$ first sorts of the sort specification.
The overall recursive construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
@ -200,7 +192,7 @@
% 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 $\CC_0 = 1$.
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 category with one object and one morphism (i.e. the terminal element of $\Cat$). $\lambda. \star$ is the terminal morphism of this object, and its right adjoint sends the only object of $1$ to the terminal object of the category $\TSet$.
The functors $R_{i-1}^i$ are the forgetful monadic functors that forget about the $i$-th sort contsructor. They have a left adjoint denoted $L_{i-1}^i$.
As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will use the two following adjunctions
@ -217,12 +209,14 @@
We construct the category $\CC_i$ as the following pair:
\[
\CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/H_i\right) = (X : \CC_{i-1}) \times \left(\Set^{H_i}\right)
\CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/H_i(X)\right) = (X : \CC_{i-1}) \times \left(\Set^{H_i(X)}\right)
\]
and where $H_i$ is a specific representable functor $\CC_{i-1} \to \Set$, such that $H_i(X)$ is the set of parameters for the construction of the new sort.
and where $H_i$ is a specific functor $\CC_{i-1} \to \Set$, such that $H_i(X)$ is the set of parameters for the construction of the new sort.
\paragraph{$H_i$ functors for Type Theory}
I will give now an example of those $H_i$ objects for our type theory example. We begin with
\todo{Do we need this functor to be representable. If so, precise it}
\paragraph{$H_i$ functors for our Type Theory example}
Let us give an example of those $H_i$ objects for our type theory example. We begin with
\[
H_1(\star) = 1 \in \operatorname{Obj}(\Set)
\]
@ -786,6 +780,32 @@
\label{apx:FG-refl}
\todo{La preuve :/}
We want to find, for each object $(X,(B,g))$ of $\CC_i = (X : \CC_{i-1}) \times (\Set/H_i(X))$, an isomorphism $(X,(B,g)) \to F_iG_i(X,(B,g))$. $g$ is a morphism from $B$ to $H_i(X)$
\[\begin{array}{rcl}
F_iG_i(X,\Rtsc)
&=& F_iW_i(G_{i-1}X,\Rtsc)\\
&=& F_i(G_{i-1}X \oplus L_0^{i-1}H_\bullet(G_{i-1}X,\Rtsc),\widetilde{\inj_2})\\
&=&(F_{i-1} \times \id)\left(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,\Rtsc),(A,h)\right)\\
&=&\left(F_{i-1}(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,\Rtsc)),(A,h)\right)
\end{array}\]
Where $(A,h)$ is the pullback defined as
\begin{center}
% YADE DIAGRAM E1.json
% GENERATED LATEX
\input{graphs/E1.tex}
% END OF GENERATED LATEX
\end{center}
We can extend this pullback using the two isomorphisms given by the induction hypothesis and hypothesis H3.
\begin{center}
% YADE DIAGRAM E2.json
% GENERATED LATEX
\input{graphs/E2.tex}
% END OF GENERATED LATEX
\end{center}
\end{document}
@ -814,6 +834,15 @@

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{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\CC{{\\ensuremath{\\mathcal{C}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":8,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":true,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":2},"to":1},{"from":1,"id":9,"label":{"kind":"normal","label":"h","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":1,"id":10,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":3,"id":11,"label":{"kind":"normal","label":"(\\en_0^i)^{-1}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":4},{"from":3,"id":12,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":5},{"from":4,"id":13,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"blue","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":6},{"from":5,"id":14,"label":{"kind":"normal","label":"(\\en_0^i)^{-1}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":6},{"from":2,"id":15,"label":{"kind":"normal","label":"\\widetilde{inj_2}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":5},{"from":0,"id":16,"label":{"kind":"normal","label":"h'","style":{"alignment":"left","bend":0,"color":"blue","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":7},{"from":7,"id":17,"label":{"kind":"normal","label":"H_iF_{i-1}(\\inj_1^i)","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"double","position":0.5,"tail":"none"},"zindex":-3},"to":2},{"from":7,"id":18,"label":{"kind":"normal","label":"\\inj_2","style":{"alignment":"left","bend":0,"color":"blue","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":6},{"from":0,"id":19,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"blue","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":-1},"to":4},{"from":9,"id":20,"label":{"kind":"pullshout","label":"","style":{"alignment":"","bend":0,"color":"black","dashed":false,"head":"","kind":"normal","position":0,"tail":""},"zindex":0},"to":10},{"from":16,"id":21,"label":{"kind":"pullshout","label":"","style":{"alignment":"","bend":0,"color":"black","dashed":false,"head":"","kind":"normal","position":0,"tail":""},"zindex":0},"to":19}],"nodes":[{"id":0,"label":{"isMath":true,"label":"A'","pos":[100,100],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"A","pos":[300,164],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"H_iF_i(G_{i-1}X \\oplus L_0^{i-1}\\Hbar(\\bullet))","pos":[300,293],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"R_0^{i-1}(G_{i-1}X\\oplus L_0^{i-1}\\Hbar(\\bullet))_\\El","pos":[596,164],"zindex":0}},{"id":4,"label":{"isMath":true,"label":"(R_0^{i-1}G_{i-1}X)_\\El \\oplus \\Hbar(\\bullet)_\\El","pos":[813,100],"zindex":0}},{"id":5,"label":{"isMath":true,"label":"R_0^{i-1}(G_{i-1}X\\oplus L_0^{i-1}\\Hbar(\\bullet))_\\UU","pos":[596,293],"zindex":0}},{"id":6,"label":{"isMath":true,"label":"(R_0^{i-1}G_{i-1}X)_\\UU \\oplus \\Hbar(\\bullet)_\\UU","pos":[813,373],"zindex":0}},{"id":7,"label":{"isMath":true,"label":"H_iF_{i-1}G_{i-1}X","pos":[100,373],"zindex":-10000}}],"sizeGrid":200,"title":"1"}]},"version":12}

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@ -2,6 +2,7 @@
\usepackage[utf8]{inputenc}
\usepackage{ae}
\usepackage[T1]{fontenc}
\usepackage[USenglish]{babel}
\usepackage{fontspec}
\usepackage{alphabeta}
\usepackage{polyglossia}
@ -72,21 +73,12 @@
% Création des environnement globaux
\newtheorem{theorem}{Théorème}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{property}{Propriété}
\newtheorem{property}{Property}
\newtheorem{remark}{Remark}
\newcounter{rule}
\addto\extrasfrench{%
\renewcommand{\figureautorefname}{\textsc{Figure}}
\renewcommand{\sectionautorefname}{Section}
\renewcommand{\subsectionautorefname}{Sous-section}
\renewcommand{\appendixautorefname}{Annexe}
\renewcommand{\theoremautorefname}{Théorème}
\providecommand\propertyautorefname{Propriété}
\providecommand\ruleautorefname{règle}
}
% Commandes logiques globales
\newcommand{\ifnullthenelse}[3]{
@ -116,6 +108,7 @@
\newcommand\Ty{{\ensuremath{\operatorname{Ty}}}}
\newcommand\Tm{{\ensuremath{\operatorname{Tm}}}}
\newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}
\newcommand\Rtsc{{\ensuremath{\operatorname{\mathcal{R}tsc}}}}
\newcommand\Cat{{\ensuremath{\operatorname{\mathcal{C}at}}}}
\newcommand\Set{{\ensuremath{\operatorname{\mathcal{S}et}}}}
\newcommand\FamSet{{\ensuremath{\operatorname{\mathcal{F}am\mathcal{S}et}}}}