Mysaa/M2Internship
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Prise en compte des commentaires, déplacement de la preuve de l'adjonction en annexe

Browse Source
This commit is contained in:
Samy Avrillon 2024-08-17 01:43:49 +02:00
parent 4effe18e0f
commit f6bc7f8db9
Signed by: Mysaa
GPG Key ID: 0220AC4A3D6A328B
1 changed files with 87 additions and 88 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

175
Report/M2Report.tex
View File

@ -26,7 +26,7 @@
\section{Introduction} \section{Introduction}
A Generalized Algebraic Theory (or GAT), first introduced by Cartmell \cite{CartmellGATs}, is a syntactic specification of an algebraic structure. From a GAT, one can define a category of models describing the models of the algebraic structure. A Generalized Algebraic Theory (or GAT), first introduced by Cartmell \cite{CartmellGATs}, is a syntactic specification of an algebraic structure. From a GAT, one can define a category of models describing the models of the algebraic structure.
A GAT typically starts with a sort specification i.e. a list of sort declarations, eventually followed by a list of constructors. A GAT typically starts with a sort specification i.e. a list of sort declarations, which may be followed by a list of constructors.
In this document, we will not ask ourselves about the specific syntax of GATs. A \enquote{vague} definition is enough to understand the rest of the document. In this document, we will not ask ourselves about the specific syntax of GATs. A \enquote{vague} definition is enough to understand the rest of the document.
\paragraph{Sort specification} \paragraph{Sort specification}
@ -46,9 +46,9 @@
A sort specification therefore specifies the differents families of sets contained in a model, and how they relate to each other in terms of indexing. A sort specification therefore specifies the differents families of sets contained in a model, and how they relate to each other in terms of indexing.
\paragraph{Constructor specification} \paragraph{Term specification}
We can also add constructors to a sort specification. They are composed of parameters (the same kind as sort declarations) and of a codomain which is a sort defined in a previous sort declaration. Those constructors specify elements of the sets contained in the model. Once we have a sort specification, we can add constructors to it in order to make a complete GAT. Constructors are composed of parameters (the same kind as for sort declarations) and of a codomain which is a sort defined by a previous sort declaration. Those constructors specify elements of the sets contained in the model.
For example, for the previous sort specification, one can add the following constructors: For example, to the previous sort specification, one can add the following constructors:
\vspace{1em} \vspace{1em}
\renewcommand\arraystretch{1.5} \renewcommand\arraystretch{1.5}
@ -61,7 +61,7 @@
\paragraph{Two-sortification} \paragraph{Two-sortification}
It was observed \cite{AmbrusSzumiXie2sort} that one can transform any GAT into a GAT with only two sorts. We will present this transformation. It was observed \cite{AmbrusSzumiXie2sort} that one can transform any GAT into a GAT with only two sorts. Let us present this transformation.
The sort specification of the transformed GAT is always the same, and contains these two sort declarations: The sort specification of the transformed GAT is always the same, and contains these two sort declarations:
@ -74,7 +74,7 @@
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: 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 GAT of Type Theory presented above becomes that which follows:
\begin{tabular}{p{0.4\textwidth}|p{0.5\textwidth}} \begin{tabular}{p{0.4\textwidth}|p{0.5\textwidth}}
$\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\ $\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\
@ -99,7 +99,8 @@
The goal of my internship was to study and understand the relationship between the categories of models of an original GAT and the category of models of the transformed \enquote{two-sortified} GAT, in order to legitimate this transformation. In this document, we will only study sort specifications (i.e. lists of sort declaration, with no constructors). The goal of my internship was to study and understand the relationship between the categories of models of an original GAT and the category of models of the transformed \enquote{two-sortified} GAT, in order to legitimate this transformation. In this document, we will only study sort specifications (i.e. lists of sort declaration, with no constructors).
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 constructed a coreflection between those two categories, whose formal definition is given in \autoref{sec:proofSection}. 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.
This coreflection justifies the transformation as the initial object of $\CC$ can be computed as the image by $F$ of the initial object of $\BB$.
The category $\BB$ is equipped with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$. The category $\BB$ is equipped with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$.
@ -113,7 +114,7 @@
\section{Examples} \section{Examples}
\label{sec:Examples} \label{sec:Examples}
Before making the formal proof, we will study our construction on three examples. Before making the formal proof, we will study our construction on three examples, in order to ease understanding of the formal proof.
\paragraph{Structure of the proof} \paragraph{Structure of the proof}
The formal 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}). The formal 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}).
@ -156,7 +157,7 @@
\end{center} \end{center}
\vspace{.5ex} \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: 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 objects and one non-trivial arrow between them. The objects and morphism of $\TT$ are described below:
\begin{center} \begin{center}
% YADE DIAGRAM CategoryTT.json % YADE DIAGRAM CategoryTT.json
@ -495,7 +496,7 @@
\todo{L'exemple, ou sinon remplacer toutes les occurences de «trois» par «deux»} \todo{L'exemple, ou sinon remplacer toutes les occurences de «trois» par «deux»}
\section{Constructing the coreflection} \section{Constructing the coreflection}
\label{sec:proofSection}
The proof will take form of an induction on the number of sorts taken into account. At each induction step, we will build the following objects: The proof will take form of an induction on the number of sorts taken into account. At each induction step, we will build the following objects:
\begin{itemize} \begin{itemize}
@ -543,14 +544,14 @@
\subsection{Preliminaries} \subsection{Preliminaries}
\paragraph{Grothendieck Construction} \paragraph{Grothendieck construction}
For a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \Cat$, the Grothendieck construction of $F$ is a category whose objects are pairs of For a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \Cat$, the Grothendieck construction of $F$ is a category whose objects are pairs of
\begin{itemize} \begin{itemize}
\item $X$ an object of $\mathcal{C}$ \item $X$ an object of $\mathcal{C}$
\item an object of $F(X)$ \item an object of $F(X)$
\end{itemize} \end{itemize}
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 $H(X')$. A morphism $(X,Y) \to (X',Y')$ is 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. We will denote this category $(X : \mathcal{C}) \times F(X)$ as its objects are pairs.
@ -576,7 +577,7 @@
This equivalence is described by the following correspondence of objects. This equivalence is described by the following correspondence of objects.
An object $(Y,f)$ of $\Set/X$ is transformed into the family of sets $A : X \to \Set$ defined by An object $(Y,f)$ of $\Set/X$ is transformed into the family of sets $A : X \to \Set$ defined by
\[ \[
A(x) := f^{-1}(\{x\}) = \text{the coimage by f of the singleton $\{x\}$} A(x) := f^{-1}(\{x\}) = \text{the preimage by f of the singleton $\{x\}$}
\] \]
Conversly, a familiy of sets $A : X \to \Set$ is transformed into the following object of $\Set/X$ Conversly, a familiy of sets $A : X \to \Set$ is transformed into the following object of $\Set/X$
@ -601,7 +602,7 @@
% END OF GENERATED LATEX % END OF GENERATED LATEX
\end{center} \end{center}
This category is equivalent to the catgegory $\FamSet$, as $(X : \Set) \times \Set^X$ is equivalent to $(X : \Set) \times (\Set/X)$ (as seen above), which is isomorphic to $\TSet$ (both categories consists of two sets and one arrow between them). This category is equivalent to the category $\FamSet$, as $(X : \Set) \times \Set^X$ is equivalent to $(X : \Set) \times (\Set/X)$ (as seen above), which is isomorphic to $\TSet$ (both categories consists of two sets and one arrow between them).
The categories of models of the transformed GATs will be built atop of this category $\BB_0 = \TSet$. The categories of models of the transformed GATs will be built atop of this category $\BB_0 = \TSet$.
@ -772,9 +773,11 @@
% END OF GENERATED LATEX % END OF GENERATED LATEX
\end{center} \end{center}
\subsection{Inductive Step: Proof of the adjunction} \subsection{Inductive Step: Proof of the coreflection}
In this subsection, we prove that $(W,E)$ make an adjunction by showing that there is a natural isomorphism between $\Hom$ sets in both categories. In this subsection, we prove that $(W,E)$ make up a coreflection i.e. an adjunction where the left adjoint is full and faithful.
\paragraph{Adjunction} We will prove the adjunction by showing that there is a natural isomorphism between $\Hom$ sets in both categories.
We want to construct for each $(X,Y)$ in $(X : \BB_{i-1}) \times (\Set/H_iF_{i-1}X)$ and $Z$ in $\BB_i$, a natural isomorphism $\phi_{XYZ}$. We want to construct for each $(X,Y)$ in $(X : \BB_{i-1}) \times (\Set/H_iF_{i-1}X)$ and $Z$ in $\BB_i$, a natural isomorphism $\phi_{XYZ}$.
@ -782,80 +785,21 @@
\phi_{XYZ} : \Hom(W(X,Y),Z) \to \Hom((X,Y),E(Z)) \phi_{XYZ} : \Hom(W(X,Y),Z) \to \Hom((X,Y),E(Z))
\] \]
I will present how the isomorphisms and its inverse are constructed. The proof that $\phi_{XYZ}$ and $\phi_{XYZ}^{-1}$ are inverse one of the other, and that they are natural are given in \autoref{apx:phi-WE-isnat}. You can find the proof of the adjunction in \autoref{apx:adjunction}, made of the following parts:
\begin{itemize}
\item The construction of $\phi_{XYZ}$
\item The construction of $\phi_{XYZ}^{-1}$
\item A proof that $\phi_{XYZ}^{-1} \circ \phi_{XYZ} (f) = f$
\item A proof that $\phi_{XYZ} \circ \phi_{XYZ}^{-1} (g,h) = (g,h)$
\item A proof that $\phi_{XYZ}$ is natural.
\end{itemize}
\subsubsection{Constructing $\phi_{XYZ}$} All these steps give us that $F_i$ and $G_i$ are in an adjunction $F_i \vdash G_i$.
Let $f$ be in $\Hom(W(X,Y),Z)$. \paragraph{Coreflection}
We want to construct $\phi_{XYZ}(f) : (X,Y) \to E(Z)$. Next, we have proven that this newly created adjunction $F_i \vdash G_i$ create a coreflection. It means that $F_iG_i \cong \Id_{\CC_i}$, or equivalently that $G_i$ is full and faithful.
The first component of $\phi_{XYZ}(f)$ is defined as the following composition: The proof of that second statement is given in \autoref{apx:FG-refl}.
\begin{center}
% YADE DIAGRAM PhiXYZFirstComponent.json
% GENERATED LATEX
\input{graphs/PhiXYZFirstComponent.tex}
% END OF GENERATED LATEX
\end{center}
The second component is defined through the universal property of the pullback defined by $E(Z)$ according to the following diagram:
\begin{center}
% YADE DIAGRAM PhiXYZSndComponentPullback.json
% GENERATED LATEX
\input{graphs/PhiXYZSndComponentPullback.tex}
% END OF GENERATED LATEX
\end{center}
\subsubsection{Constructing $\phi^{-1}_{XYZ}$}
Now, we take $(g,h)$ a morphism from $(X,Y)$ to $E(Z)$.
The morphism $\phi^{-1}_{XYZ}(g,h)$ of $\BB_i$ from $W(X,Y)$ to $Z$ is a morphism of $\BB_{i-1}$ from $X \tl^{i-1} K_{H_iF_{i-1}} (X,Y)$ to $R_{i-1}^i(Z)$ that make a specific diagram commute. We build this morphism using the universal property of $\tl^{i-1}$, stated in the introduction of this section.
\[
\phi^{-1}_{XYZ}(g,h) := \left\{g ; \square \right\}
\]
\todo{Est-ce qu'il faut que j'écrive $R_{i-1}^i\phi^{-1}_{XYZ}(g,h) := ...$ pour être plus «homogène» ?}
Where $\square$ is a morphism $K_{H_iF_{i-1}} (X,Y) \to R_0^i Z$ in $\TSet$ defined by the following diagram:
\begin{center}
% YADE DIAGRAM PhiXYZ-1Square.json
% GENERATED LATEX
\input{graphs/PhiXYZ-1Square.tex}
% END OF GENERATED LATEX
\end{center}
Finally, we check that $\phi^{-1}_{XYZ}(g,h)$ is a morphism of $\BB_i$ from $W(X,Y)$ to $Z$, i.e. that it makes the following diagram commute:
\begin{center}
% YADE DIAGRAM PhiXYZ-1MorphismOfBi.json
% GENERATED LATEX
\input{graphs/PhiXYZ-1MorphismOfBi.tex}
% END OF GENERATED LATEX
\end{center}
In order to complete this proof, we need to show that
\begin{itemize}
\item $\phi_{XYZ}^{-1} \circ \phi_{XYZ} (f) = f$
\item $\phi_{XYZ} \circ \phi_{XYZ}^{-1} (g,h) = (g,h)$
\item $\phi_{XYZ}$ is natural.
\end{itemize}
The proofs of these statements are given in \autoref{apx:phi-WE-isnat}.
\subsubsection{Coreflection}
We have proven that this newly created adjunction $F_i \vdash G_i$ create a coreflection. It means that $F_iG_i \cong \Id_{\CC_i}$, or equivalently that $G_i$ is full and faithful.
The proof is that statement is given in \autoref{apx:FG-refl}.
\subsection{Inductive step: Constructing $\tl^i$} \subsection{Inductive step: Constructing $\tl^i$}
@ -1166,8 +1110,63 @@
\appendixpage \appendixpage
\section{$W \dashv E$ adjunction} \section{$W \dashv E$ adjunction}
\label{apx:phi-WE-isnat} \label{apx:adjunction}
\subsubsection{Constructing $\phi_{XYZ}$}
Let $f$ be in $\Hom(W(X,Y),Z)$.
We want to construct $\phi_{XYZ}(f) : (X,Y) \to E(Z)$.
The first component of $\phi_{XYZ}(f)$ is defined as the following composition:
\begin{center}
% YADE DIAGRAM PhiXYZFirstComponent.json
% GENERATED LATEX
\input{graphs/PhiXYZFirstComponent.tex}
% END OF GENERATED LATEX
\end{center}
The second component is defined through the universal property of the pullback defined by $E(Z)$ according to the following diagram:
\begin{center}
% YADE DIAGRAM PhiXYZSndComponentPullback.json
% GENERATED LATEX
\input{graphs/PhiXYZSndComponentPullback.tex}
% END OF GENERATED LATEX
\end{center}
\subsubsection{Constructing $\phi^{-1}_{XYZ}$}
Now, we take $(g,h)$ a morphism from $(X,Y)$ to $E(Z)$.
The morphism $\phi^{-1}_{XYZ}(g,h)$ of $\BB_i$ from $W(X,Y)$ to $Z$ is a morphism of $\BB_{i-1}$ from $X \tl^{i-1} K_{H_iF_{i-1}} (X,Y)$ to $R_{i-1}^i(Z)$ that make a specific diagram commute. We build this morphism using the universal property of $\tl^{i-1}$, stated in the introduction of this section.
\[
\phi^{-1}_{XYZ}(g,h) := \left\{g ; \square \right\}
\]
\todo{Est-ce qu'il faut que j'écrive $R_{i-1}^i\phi^{-1}_{XYZ}(g,h) := ...$ pour être plus «homogène» ?}
Where $\square$ is a morphism $K_{H_iF_{i-1}} (X,Y) \to R_0^i Z$ in $\TSet$ defined by the following diagram:
\begin{center}
% YADE DIAGRAM PhiXYZ-1Square.json
% GENERATED LATEX
\input{graphs/PhiXYZ-1Square.tex}
% END OF GENERATED LATEX
\end{center}
Finally, we check that $\phi^{-1}_{XYZ}(g,h)$ is a morphism of $\BB_i$ from $W(X,Y)$ to $Z$, i.e. that it makes the following diagram commute:
\begin{center}
% YADE DIAGRAM PhiXYZ-1MorphismOfBi.json
% GENERATED LATEX
\input{graphs/PhiXYZ-1MorphismOfBi.tex}
% END OF GENERATED LATEX
\end{center}
\subsection{Composition $\phi_{XYZ} \circ \phi_{XYZ}^{-1}$} \subsection{Composition $\phi_{XYZ} \circ \phi_{XYZ}^{-1}$}