Modifications du rapport selon les retours. Suppression d'une bonne partie des TODO
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@ -33,7 +33,7 @@
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A sort specification is a list of \emph{sort declarations}, composed of \emph{parameters} and $\Set$ as their codomain.
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We give an example of a sort specification for Type Theory. On the right column we give the interpretation of the sort declarations as components of the models.
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We give an example of a sort specification for Type Theory. On the right column we give the interpretation of the sort declarations as components of the models of the GAT.
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\vspace{1em}
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\renewcommand\arraystretch{1.5}
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@ -44,10 +44,10 @@
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\end{tabular}
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\vspace{1em}
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A sort specification therefore specifies the differents kinds of sets contained in a model, and how they relate to each other.
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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.
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\paragraph{Constructor specification}
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We can also add constructors to a sort specification. They are composed as parameters (the same kind as sort declarations) and a codomain which is a sort defined in a previous sort declaration. Those constructors specify elements of the sets contained in the model, previously defined by the sort declaration.
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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.
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For example, for the previous sort specification, one can add the following constructors:
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\vspace{1em}
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@ -89,9 +89,7 @@
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$\dots$
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\end{tabular}
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This process is useful when studying GATs, as it allows to restrict a study to GATs with only two sorts without loss of generality. It has been used in Philippo Sestini's thesis, although it has not been formally justified:
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\todo{Ça veut dire qqch que le \enquote{process} pas été \enquote{justified} ?}
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This process is useful when studying GATs, as it allows to restrict a study to GATs with only two sorts without loss of generality. Filippo Sestini noticed that in his thesis \cite{SestiniPhD}, although they didn't prove it.
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\begin{quote}
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Many instances of multi-sorted IITs [IITs are a variant 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.
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@ -99,13 +97,11 @@
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\paragraph{Goal}
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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.
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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).
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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.
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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.
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\todo{J'ai toujours pas regardé les terms constructors, je peux le mettre, mais je ne serais pas sûr de pouvoir le défendre}
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The category $\BB$ is equipped with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$.
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\begin{center}
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% YADE DIAGRAM G1-0.json
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@ -118,22 +114,22 @@
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\paragraph{Structure of the proof}
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We will only only formalize the transformation for \emph{sort specification} (i.e. lists of sort declaration). We will show how we take into account the constructors in \autoref{sec:constructors2trans}.
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We will formalize the transformation for \emph{sort specifications} (i.e. lists of sort declarations).
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This proof is a big 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}).
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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}).
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At the $i$-th recursion step, we will build the following objects :
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\begin{itemize}
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\setlength\itemsep{-1ex}
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\item The category of models of the GAT $\CC_i$
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\item The category of models of the transformed GAT $\BB_i$
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\item A functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$, taking track of the fixed sort specification of the transformed GAT. \inlinetodo{Mal Dit, est-ce nécessaire ?}
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\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 $(\mathcal{O},\El,\Con,\Ty)$.
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Those functors compose themselves into a functor $R_0^i : \BB_i \to \BB_0$.
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Those functors compose themselves into a functor $R_0^i : \BB_i \to \BB_0$, that also creates limits:
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\[
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R_0^i = R^1_0 \circ \dots \circ R_{i-1}^i
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R_0^i := R^1_0 \circ \dots \circ R_{i-1}^i
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\]
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\item An bifunctor $\tl^i : \BB_i \times \BB_0 \to \BB_i$ along with a morphism $\inj_\tl^i : X \to X \tl^i Y$ for every $X,Y$ in $\BB_i \times \BB_0$, and an isomorphism $\en_{i-1}^i : R_{i-1}^i (X \tl^i Y) \to (R_{i-1}^i X) \tl^{i-1} Y$. This bifunctor 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:
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\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:
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\label{sec:tlUniversalProperty}
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\begin{center}
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@ -143,17 +139,19 @@
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% END OF GENERATED LATEX
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\end{center}
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where $\en_0^i$ is the composition $\en_0^1 \circ R_0^1\en_1^2 \circ \dots \circ R_0^{i-1}\en_{i-1}^i : R_0^i(X \tl^i Y) \to (R_0^iX) \oplus Y = (R_0^iX)\tl^0 Y$.
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where $\en_0^i$ is the following composition:
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We will construct later that $\tl^0$ is $\oplus$, the coproduct of the category $\BB_0 = \TSet$. The second injector of this coproduct is denoted as $\inj_2$.
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\[\en_0^i := R_0^{i-1} \en_{i-1}^i \circ \dots \circ R_0^1\en_1^2 \circ \en_0^1 : (R_0^i X) \oplus Y = (R_0^i X) \tl^0 Y \to R_0^i (X \tl^i Y)\]
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\todo{Faut-il remplacer $\en$ par $\en^{-1}$, vu qu'on utilise $\en^{-1}$ partout ?}
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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.
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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 $\BB_i$.
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\item A functor $F_i : \BB_i \to \CC_i$
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\item A functor $G_i : \CC_i \to \BB_i$
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\item An adjunction between them $F_i \vdash G_i$
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\item A proof that $F_iG_i \cong \Id_{\CC_i}$ (i.e. $F_i \vdash G_i$ make up a coreflection)
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\item A proof that $F_i\inj_\tl^i$ is an isomorphism
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\item A proof that $F_i\inj_\tl^i$ is an isomorphism. The inverse isomorphism will be denoted as $(F_i\inj_\tl^i)^{-1}$
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\end{itemize}
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Here is a figure that describes the recursive construction of some of the above objects
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\subsection{Preliminaries}
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\paragraph{Grothendieck Construction}
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For a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \Cat$, the Grothendieck construction is a category whose objects are pairs of
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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
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\begin{itemize}
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\item $X$ an object of $\mathcal{C}$
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\item an object of $F(X)$
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We will denote this category $(X : \mathcal{C}) \times F(X)$ as its objects are pairs.
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\paragraph{Slice category}
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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
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For a category $\mathcal{C}$ and $X$ an object in that category, the slice category (or over category) $\mathcal{C}/X$ is a category whose objects are pairs of
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\begin{itemize}
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\item $Y$ an object of $\mathcal{C}$
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\item an arrow $X \to Y$ of $\mathcal{C}$
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\item an arrow $Y \to X$ of $\mathcal{C}$
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\end{itemize}
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A morphism $(Y,f) \to (Y',f')$ is a morphism $g : Y \to Y'$ such that $g \circ f = f'$.
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A morphism $(Y,f) \to (Y',f')$ is a morphism $g : Y \to Y'$ such that $f' \circ g = f$.
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We can define a functor $\left(\mathcal{C}/\dash\right) : \mathcal{C} \to \Cat$ from this construction.
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If the category $\mathcal{C}$ is $\Set$, we have the equivalence $\Set/X \simeq \Set^X$.
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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 $(X : \mathcal{C}) \times \left(\Set\middle/H(X)\right)$.
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\paragraph{Category of models of the two-sort sort specification}
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\label{sec:categoryOfModelsOfTwoSorts}
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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 another category : $\TSet$, the category 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.
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\paragraph{Slice category over a set}
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When the category $\mathcal{C}$ is $\Set$, we have the equivalence
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\[\Set/X \simeq \Set^X\]
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This equivalence is described by the following correspondence of objects.
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An object $(Y,f)$ of $\Set/X$ is transformed into the family of sets $A : X \to \Set$ defined by
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\[
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A(x) := f^{-1}(\{x\}) = \text{the coimage by f of the singleton $\{x\}$}
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\]
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Conversly, a familiy of sets $A : X \to \Set$ is transformed into the following object of $\Set/X$
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\begin{center}
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% YADE DIAGRAM G0.json
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% YADE DIAGRAM EquivSetXSetX.json
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% GENERATED LATEX
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\input{graphs/G0.tex}
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\input{graphs/EquivSetXSetX.tex}
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% END OF GENERATED LATEX
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\end{center}
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This category is equivalent to the catgegory $\FamSet$ with an equivalence that is given in \inlinetodo{annex reference}, with the following equivalence:
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\todo{Choose whether to tell the equivalence}
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where $\coprod$ is the coproduct of $\Set$ and $\pi_1$ is the first projection of the coproduct.
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Morphisms are translated using the same logic.
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\paragraph{Category of models of the two-sort sort specification}
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\label{sec:categoryOfModelsOfTwoSorts}
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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 $\FamSet$, defined as $(X:\Set) \times \Set^X$. 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:
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\begin{center}
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% YADE DIAGRAM CategoryTT.json
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% GENERATED LATEX
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\input{graphs/CategoryTT.tex}
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% END OF GENERATED LATEX
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\end{center}
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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).
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With this formalisation, a model of the two-sort GAT is a functor $X : \TSet$, such that
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\begin{itemize}
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\item $X_\UU$ is the set of the \enquote{sort objects}
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\item For each sort object $\Gamma \in X_\UU$, the set of objects corresponding to the sort object is $X_p^-1(\{\Gamma\}) \subset X_\El$
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\item For each sort object $\Gamma \in X_\UU$, the set of objects corresponding to the sort object is $X_p^{-1}(\{\Gamma\}) \subset X_\El$
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\end{itemize}
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Therefore the categories of models of the transformed GATs will be built atop of this category $\BB_0 = \TSet$.
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\paragraph{$K$ functor}
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We define a functor $K : (X:C) \times (\Set/A(X)) \to \TSet$ defined as follows
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We define a helper functor $K : (X:C) \times (\Set/A(X)) \to \TSet$ defined as follows
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\[
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K_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)}
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\]
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\end{remark}
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\subsection{Initialization}
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In this subsection, we will build the objects of the first step of our induction. This will correspond to the empty sort specification.
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In this subsection, we will build the objects of the first step of our induction. This will correspond to the empty sort specification, i.e. the induction step $i = 0$.
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\begin{itemize}
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\setlength\itemsep{-1ex}
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\item $\CC_i$ is $\one$, the category with only one object and one trivial morphism (i.e. the terminal category of $\Cat$), because the empty sort specification only has one model. \inlinetodo{Isn't it up to isomorphism ?}
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\item $\BB_i$ is $\TSet$, the category of sorts of the $(\mathcal{O},\El)$ sort specification (see \autoref{sec:categoryOfModelsOfTwoSorts} for its description)
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\item $\tl^0 : \BB_0 \times \BB_0 \to \BB_0$ is the coproduct $\oplus$ of $\TSet$, with $\inj_\tl^0 : X \to X \oplus Y$ being the first injector of the coproduct. We will also denote as $\inj_2^0$ the second injector of this coproduct. Please note that there is only a second injector for the initial step of the recursion.
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The universal property of $\tl^0$ is that of the coproduct.
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\todo{Est-ce que ça pose problème de parler de \emph{the} coproduct ?}
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\item $\CC_0$ is $\one$, the category with only one object $\star$ and one trivial morphism (i.e. the terminal category of $\Cat$), because the empty sort specification only has one model.
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\item $\BB_0$ is $\TSet$, the category of models of the $(\mathcal{O},\El)$ sort specification (see dedicated paragraph in last subsection for a description).
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\item The universal property of $\tl^0$ is that of the coproduct. Therefore, we will define $\tl^0 : \BB_0 \times \BB_0 \to \BB_0$ to be the coproduct $\oplus$ of $\TSet$, with $\inj_\tl^0 : X \to X \oplus Y$ being the first injector of the coproduct. The second injector of the coproduct $\oplus$ will be refered as $\inj_2$.
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\item $F_0 : \TSet \to \one$ is the terminal functor of $\Cat$
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\item $G_0 : \one \to \TSet$ is the functor that sends the only object of $\one$ to the initial object of $\TSet$: $0_\TSet$
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\item With $\star$ being the only object of $\one$, and for $X$ an object of $\TSet$, we have
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\[\Hom(G_0 \star,X) = \Hom(0_\TSet,X) \cong 1 \cong \Hom(\star,F_0 X)\]
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\item For $X$ an object of $\TSet$, we have
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\[\Hom(G_0 \star,X) = \Hom(0_\TSet,X) \cong \one \cong \Hom(\star,F_0 X)\]
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(reminder: $\star$ is the only object of $\one$)
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Therefore, we have that $F_0 \vdash G_0$.
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\todo{Are the iso/equiv equalities correct ?}
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\item $F_0G_0 : \one \to \one$ so $F_0G_0 = \Id_\one$ as $\one$ is terminal in $\Cat$
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\item $F_i\inj_1 = \id_\star$ which is an isomorphism
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\end{itemize}
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\subsection{Constructing the categories}
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\subsection{Inductive Step: Constructing the categories}
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\label{sec:constructingCategory}
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In this part, i will show how we construct recursively both categories $\CC_i$ and $\BB_i$, along with the functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$. We will use the loop invariants that we expressed in the introduction of this section.
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In order to construct the categories, we need some object describing the specific sort we are adding to the categories. This object is a specific functor $H_i : \CC_{i-1} \to \Set$. We suppose that those $H_i$ functors are given.
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In order to construct the categories, we need some object describing the specific sort we are adding to the categories. This object is a specific functor $H_i : \CC_{i-1} \to \Set$. Those functors are the same that Altenkirsh and al. \cite{Altenkirch2018} used in their paper to describe a specification of sorts.
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We suppose that those $H_i$ functors are given.
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\begin{remark}
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There is a way of getting the functor $H_i$ from the syntax, which is described in \autoref{sec:HiFromSyntax}.
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\]
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and where $H_i$ is the specific functor described above.
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\todo{Do we need this functor to be representable ? If so, precise it}
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\paragraph{An example of $H_i$ functors}
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Let us now give an example of those $H_i$ objects for our type theory example.
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\]
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We begin with the following functor, corresponding to the sort declaration above
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\[
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H_1(\star) = 1 \in \operatorname{Obj}(\Set)
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H_1(\star) = 1 \in \Set
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\]
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This corresponds to the fact that $\Con$ takes no parameter.
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Therefore $\CC_1 = 1 \times \Set^1 = \Set$, which is as expected: a model is a set.
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Therefore $\CC_1 = 1 \times \Set^1 \simeq \Set$, which is as expected: a model $X$ consists of a set $X_\Con$.
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\[
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\boxed{\Ty : (\Gamma : \Con) \to \Set}
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\]
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Then, we take the functor $H_2(X_\Con) = X_\Con$, corresponding to the sort declaration above. This functor means that types need \emph{one} context to be built.
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Then, we take the functor $H_2(X) = X_\Con$, corresponding to the sort declaration above. This functor means that types need \emph{one} context to be built.
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Therefore $\CC_2 = (X:\Set) \times (\Set/X) \simeq (X:\Set) \times (\Set^X) = \FamSet$, a model is a family of sets, as expected.
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Therefore $\CC_2 = (X:\Set) \times (\Set/X) \simeq (X:\Set) \times (\Set^X) = \FamSet$, a model $X$ is a family of sets $\left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con}$, as expected.
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\[
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\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}
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\]
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Finally, we take the functor $H_3(X_\Con,X_\Ty) = \sum_{\Gamma : X_\Con}X_\Ty(\Gamma)$. It means that terms need \emph{one} context, and \emph{one} type of that context.
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Finally, we take the functor $H_3(X) = \sum_{\Gamma : X_\Con}X_\Ty(\Gamma)$. It means that terms need \emph{one} context, and \emph{one} type of that context.
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An object of that final category $\CC_3$ is a triple
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\[
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@ -307,7 +320,7 @@
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We construct the category $\BB_i$ as follows.
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An object of $\BB_i$ is
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An object of $\BB_i$ is a pair of:
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\begin{itemize}
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\item an object $X$ of $\BB_{i-1}$
|
||||
\item a \enquote{sort constructor} $\Cstr^X$ as a function $H_iF_{i-1}X \to (R_0^{i-1}X)_\UU$
|
||||
@ -315,7 +328,7 @@
|
||||
where $H_i$ is the functor $\CC_{i-1} \to \Set$ described above and $F_{i-1}$ is the right part of the adjunction $\BB_{i-1} \to \CC_{i-1}$ that we have defined at the previous induction step.
|
||||
\end{itemize}
|
||||
|
||||
If we have an object $X$ of $\BB_i$, the first component is denoted as $R_{i-1}^iX \in \operatorname{Obj}(\BB_{i-1})$, and the second component is denoted as $\Cstr^X : R_{i-1}^i X \to (R_0^iX)_\UU$.
|
||||
If we have an object $X$ of $\BB_i$, the first component is denoted as $R_{i-1}^iX$, which is an object of $\BB_{i-1}$, and the second component is denoted as $\Cstr^X : R_{i-1}^i X \to (R_0^iX)_\UU$.
|
||||
|
||||
A morphism $X \to X'$ of $\BB_i$ is a morphism $f : R_{i-1}^iX \to R_{i-1}^iX'$ in $\BB_{i-1}$ such that the following diagram commutes.
|
||||
|
||||
@ -330,9 +343,9 @@
|
||||
|
||||
Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
|
||||
|
||||
$R_{i-1}^i$ acts on objects and morphisms, and therefore creates a functor $\BB_i \to \BB_{i-1}$.
|
||||
$R_{i-1}^i$ acts on objects and morphisms, and induces a functor $\BB_i \to \BB_{i-1}$. The induced functor is monadic, and therefore preserves limits.
|
||||
|
||||
\subsection{Constructing the functors}
|
||||
\subsection{Inductive step: Constructing the functors $F_i$ and $G_i$}
|
||||
|
||||
The adjunction $F_i \vdash G_i$ is built using the two functors from the adjunction $F_{i-1} \vdash G_{i-1}$ defined in the previous induction step. We use them to define the first part of the adjunction, and we compose them with two adjoint functors $W$ and $E$ that we will define in this section. The overall construction for this induction step is as follows:
|
||||
|
||||
@ -357,12 +370,12 @@
|
||||
\[
|
||||
(G_{i-1} \times (H_i\eta_{i-1} \circ \dash))(f,g) = (G_{i-1}f,g)
|
||||
\]
|
||||
$\eta_{i-1} : X \to F_{i-1}G_{i-1}X$ denotes the unit of the adjunction $F_{i-1} \vdash G_{i-1}$.
|
||||
with $\eta_{i-1} : X \to F_{i-1}G_{i-1}X$ being the unit of the adjunction $F_{i-1} \vdash G_{i-1}$.
|
||||
|
||||
These equations define two functors, that create an adjunction, as $F_{i-1} \vdash G_{i-1}$ is an adjuntion, and the action on morphism is that of $F_{i-1}$ and $G_{i-1}$.
|
||||
|
||||
\todo{Est-ce que c'est clair ? Besoin de plus de preuve ? Prouver que c'est des foncteurs ? Une adjonction ?}
|
||||
These equations define two functors, that create the following adjunction:
|
||||
\[F_{i-1} \times \id \vdash G_{i-1} \times (H_i\eta_{i-1} \circ \dash)\]
|
||||
|
||||
The unit of this adjunction is $\eta_{i-1} \times (H_i\eta_{i-1}\circ \dash)$ and its counit is $\varepsilon_{i-1} \times \id$, where $\varepsilon_{i-1}$ is the counit of the adjunction $F_{i-1} \vdash G_{i-1}$
|
||||
|
||||
\subsubsection{W definition}
|
||||
|
||||
@ -395,8 +408,6 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
\todo{Fixer l'affichage de ce diagramme}
|
||||
|
||||
\subsubsection{E definition}
|
||||
|
||||
We define $E : \BB_{i} \to \left(X : \BB_{i-1}\right) \times (\Set/H_iF_{i-1}X)$
|
||||
@ -414,8 +425,6 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
\todo{Faut justifier qu'on puisse faire un pullback, non ? C'est quoi l'explication la moins complexe ?}
|
||||
|
||||
The action on a morphism $f$ from $X$ to $X'$ is defined as $E(f) = (R_{i-1}^i f, !)$, with $!$ being the only morphism making the following diagram commute (thanks to the pullback property):
|
||||
|
||||
\begin{center}
|
||||
@ -425,7 +434,7 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
\subsection{Proof of the adjunction}
|
||||
\subsection{Inductive Step: Proof of the adjunction}
|
||||
|
||||
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.
|
||||
|
||||
@ -470,6 +479,8 @@
|
||||
\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}
|
||||
@ -501,17 +512,16 @@
|
||||
|
||||
\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 $G_i is full and faithful$.
|
||||
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{Other objects}
|
||||
\subsubsection{Constructing $\tl^i$}
|
||||
\subsection{Inductive step: $\tl^i$}
|
||||
|
||||
\label{sec:coproductConstr}
|
||||
|
||||
\paragraph{Constructing the objects}
|
||||
We will define the $\tl^i$ bifunctor of two objects $X$ from $\BB_i$ and $Y$ from $\BB_0$ as follows:
|
||||
We will define the $\tl^i$ operator of two objects $X$ from $\BB_i$ and $Y$ from $\BB_0$ as follows:
|
||||
\[
|
||||
X \tl_i Y := \left((R_{i-1}^i X) \tl^{i-1} Y, (R_0^{i-1} \inj_\tl^{i-1})_\UU \circ \Cstr_i^X \circ (H_iF_{i-1}\inj_\tl^{i-1})^{-1}\right)
|
||||
\]
|
||||
@ -559,7 +569,24 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
With this definition, the isomorphism $\en_{i-1}^i : R_{i-1}^i(X \tl^i Y) \to (R_{i-1}^i X) \tl^{i-1} Y$ is simply the identity morphism.
|
||||
With this definition, the isomorphism $\en_{i-1}^i : (R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i(X \tl^i Y)$ is simply the identity morphism.
|
||||
|
||||
\paragraph{Building morphisms}
|
||||
|
||||
For any two morphisms $g : X \to X'$ of $\BB_i$ and $h : Y \to Y'$ of $\BB_0$, we will create a morphism $X \tl^i Y \to X' \tl^i Y'$ as follows:
|
||||
\[
|
||||
g\tl^ih := R_{i-1}^ig \tl^{i-1} h
|
||||
\]
|
||||
\todo{Même question d'homogénéité}
|
||||
|
||||
This is indeed a morphism of $\BB_i$ as it makes the following diagram commute:
|
||||
|
||||
\begin{center}
|
||||
% YADE DIAGRAM TlDefOnMorphisms.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/TlDefOnMorphisms.tex}
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
\paragraph{Composition with $F_i$}
|
||||
We finally need to prove, for any objects $X$ in $\BB_i$ and $Y$ in $\TSet$, that the morphism
|
||||
@ -577,6 +604,8 @@
|
||||
|
||||
\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 $S$: $\left[S,\Set\right]$.
|
||||
@ -584,7 +613,7 @@
|
||||
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 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}
|
||||
|
||||
@ -650,7 +679,7 @@
|
||||
\subsection{$G$ and $F$ infinite constructions}
|
||||
|
||||
\[
|
||||
G_i(Y) = \left(\sum_{a\in S_i}H_{Y(a)}\left(\lim_{(a/S_i)*}\overline{Y_a}\right), λa.λ\eta.(a,u (\inj_2 \star))\right)
|
||||
G_i(Y) = \left(\sum_{a\in S_i}H_{Y(a)}\left(\lim_{(a/S_i)*}\overline{Y_a}\right), \lambda a.\lambda \eta.(a,u (\inj_2 \star))\right)
|
||||
\]
|
||||
|
||||
where $u : \left(Y(a) \oplus 1, \inj_1\right) \to (\lim_{(a/S_i)*} \overline{Y_a})$
|
||||
@ -661,28 +690,12 @@
|
||||
F_i(X,\Cstr)(a) = X(p)^{-1}\left(\Cstr_a\left(\Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,X)\right)\right)
|
||||
\]
|
||||
\[
|
||||
F_i(X,\Cstr)(f : a \to b)(X(p)(x); x \in \Cstr_a(\eta)) = \eta_\El^b(f)
|
||||
F_i(X,\Cstr)(f : a \to b)(X(p)(x); x \in \Cstr_a(\eta)) = \eta_\El^b(f)
|
||||
\]
|
||||
|
||||
\todo{Show that those are the same functors as those defined recursively. Prove the adjunction/reflection infinitely ?}
|
||||
|
||||
|
||||
\subsection{$H$ functors}
|
||||
|
||||
For every set $X$, we define the functor $H_X : (X/\Set) \to \TSet$
|
||||
\[
|
||||
H_X(Y,f) = \TSetObject{X}{f}{Y}
|
||||
\]
|
||||
|
||||
Dually, we make another functor $\Hbar_X : (\Set/X) \to \TSet$
|
||||
\[
|
||||
\Hbar_X(Y,f) = \TSetObject{Y}{f}{X}
|
||||
\]
|
||||
|
||||
The morphisms translate as-is, and composition and identity relies on that of $(X/\Set)$ or $(\Set/X)$.
|
||||
|
||||
\todo{(small) Show that it is actually a functor (should be trivial), potentially add a diagram}
|
||||
|
||||
\subsection{Adding 2-transformation of constructors}
|
||||
\label{sec:constructors2trans}
|
||||
\todo{Describe the process}
|
||||
@ -748,7 +761,7 @@
|
||||
|
||||
The first component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ is
|
||||
\[
|
||||
R_{i-1}^i(\left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\}) \circ \inj_1^{i-1} = \left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\} \circ \inj_1^{i-1} = g
|
||||
R_{i-1}^i(\phi_{XYZ}^{-1}(g,h)) \circ \inj_\tl^{i-1} = \left\{g ; \square \right\} \circ \inj_\tl^{i-1} = g
|
||||
\]
|
||||
|
||||
The second component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ follows the following diagram
|
||||
@ -760,24 +773,19 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
The diagram commutes as the following equality holds:
|
||||
\[
|
||||
\left(\left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\} \circ \inj^{i-1}_2\right)_\El = \left(\varepsilon_0^i \circ L_0^{i-1} \square\right)_\El = (\varepsilon_0^i)_\El \circ (L_0^{i-1} \square)_\El = \square_El = \pi_1 \circ h
|
||||
\]
|
||||
|
||||
\todo{Justify $(\varepsilon_0^i)_\El = id_{\BB_i}$ and $(L_0^i(f))_\El = f_\El$}
|
||||
\todo{Justifier que la partie du haut commute, i.e. que \[
|
||||
(R_0^{i-1}\{g,\square\})_\El \circ (\en_0^i)_\El \circ (\inj_2)_\El = \square_\El
|
||||
\]}
|
||||
|
||||
So, as the square is a pullback, we get the complete equality
|
||||
\[
|
||||
\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h)) = (g,h)
|
||||
\]
|
||||
The diagram commutes, and so we can deduce that the second component of $\phi_{XYZ}(f)$ is $h$, by proprty of the pullback $E(Z)$
|
||||
|
||||
\subsection{Composition $\phi_{XYZ}^{-1} \circ \phi_{XYZ}$}
|
||||
|
||||
Now, the converse composition is
|
||||
|
||||
\[
|
||||
\phi_{XYZ}^{-1} (\phi_{XYZ}(f)) = \left\{R_{i-1}^i f \circ \inj_1^{i-1} ; \varepsilon_0^i \circ L_0^{i-1} \square \right\}
|
||||
\phi_{XYZ}^{-1} (\phi_{XYZ}(f)) = \left\{R_{i-1}^i f \circ \inj_\tl^{i-1} ; \square \right\}
|
||||
\]
|
||||
where $\square$ follows the following diagram
|
||||
|
||||
@ -788,24 +796,15 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
We want to show that $\phi_{XYZ}^{-1} (\phi_{XYZ}(f))$. By definition of $\{;\}$ in $\BB_1$, it suffices to show that $\varepsilon_0^i \circ L_0^{i-1} \square = R_{i-1}^i f \circ \inj_1^{i-1}$.
|
||||
|
||||
So it suffices to show that
|
||||
\[
|
||||
\square = R_0^{i-1}(R_{i-1}^i f \circ \inj_2^{i-1}) \circ \eta_0^{i-1} = R_0^i f \circ \left(R_0^{i-1} \inj_2^{i-1} \circ \eta_0^{i-1}\right) = R_0^i f \circ \inj_2^0
|
||||
\]
|
||||
We want to show that $\phi_{XYZ}^{-1} (\phi_{XYZ}(f)) = f$. By definition of $\{;\}$ in $\BB_i$, it suffices to show that $\square = R_0^i f \circ \en_0^{i-1} \circ \inj_2$.
|
||||
|
||||
The two required equalities are proved by the diagram above:
|
||||
|
||||
\begin{align*}
|
||||
\square_\El = \left(R_0^i f \circ \inj_2^0\right)_\El \\
|
||||
\square_\UU = \left(R_0^i f \circ \inj_2^0\right)_\UU
|
||||
\square_\El = \left(R_0^i f \circ \en_0^{i-1} \circ \inj_2\right)_\El \\
|
||||
\square_\UU = \left(R_0^i f \circ \en_0^{i-1} \circ \inj_2\right)_\UU
|
||||
\end{align*}
|
||||
|
||||
\todo{Expliciter à un endroit que $\Cstr^{W(X,Y)} = inj_2^\Set \circ \left(\inj_1^{i-1} \circ \dash\right)$ (déduit de la définition et de la forme de l'iso H3' et H1r=id)}
|
||||
|
||||
\todo{Show $R_0^{i-1} \inj_2^{i-1} \circ \eta_0^{i-1} = \inj_2^0$}
|
||||
|
||||
\subsection{Naturality}
|
||||
|
||||
We want to show that the following diagram commutes, for any objects $X$,$Y$,$Z$,$X'$,$Y'$,$Z'$ and morphisms $f$,$g$,$h$.
|
||||
@ -817,16 +816,16 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
We take a morphism $\ii$ in $\Hom\left(W(X,Y),Z\right)$. We want to show that it is sent by the above diagram to the same morphism of $\Hom\left((X,Y),E(Z)\right)$.
|
||||
We take a morphism $\ii$ in $\Hom\left(W(X,Y),Z\right)$. We want to show that it is sent by the above diagram to the same morphism of $\Hom\left((X',Y'),E(Z')\right)$.
|
||||
|
||||
We first look at the first component of the result morphism.
|
||||
|
||||
\[\begin{array}{rcl}
|
||||
\phi_{XYZ}(f \circ \ii \circ W(g,h))_1
|
||||
&=& R_{i-1}^i\left(f \circ \ii \circ (g \oplus L_0^{i-1} \Hbar(g,h))^+\right) \circ \inj_1^{i-1} \\
|
||||
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \left\{\inj_1^{i-1} \circ g; \dots\right\} \circ \inj_1^{i-1} \\
|
||||
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \inj_1^{i-1} \circ g \\
|
||||
&=& \left[E(f) \circ \phi_{X'Y'Z'}(\ii) \circ (g,h)\right]_1
|
||||
\phi_{X'Y'Z'}(f \circ \ii \circ W(g,h))_1
|
||||
&=& R_{i-1}^i\left(f \circ \ii \circ (g \tl^{i-1} K_\bullet(g,h))^+\right) \circ \inj_\tl^{i-1} \\
|
||||
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ (g \tl^{i-1} K_\bullet(g,h)) \circ \inj_\tl^{i-1} \\
|
||||
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \inj_\tl^{i-1} \circ g \\
|
||||
&=& \left[E(f) \circ \phi_{XYZ}(\ii) \circ (g,h)\right]_1
|
||||
\end{array}\]
|
||||
|
||||
The second components are defined as described by the following diagram
|
||||
@ -838,8 +837,13 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
The second projection of $\phi_{XYZ}(f \circ \ii \circ W(g,h))$ is defined by the pullback of $E(Z')$ commuting with the two path highlighted by the blue line. And that of $E(f) \circ \phi_{X'Y'Z'}(\ii) \circ (g,h)$ is defined by the circled path.
|
||||
As the diagram commutes and by pullback property, we get the equality.
|
||||
The second projection of $\phi_{XYZ}(f \circ \ii \circ W(g,h))$ is defined by the pullback of $E(Z')$ commuting with the two path highlighted by the inner blue line. And that of $\phi_{X'Y'Z'}(\ii)$ is defined by the red outer line.
|
||||
The outer squares commute, and therefore, by the pullback property, we get that
|
||||
\[
|
||||
\phi_{X'Y'Z'}(f\circ\ii\circ W(g,h))_2 = \left[E(f) \circ \phi_{XYZ}(\ii) \circ (g,h)\right]_2
|
||||
\]
|
||||
|
||||
And thus we have naturality.
|
||||
|
||||
|
||||
\section{$F_i \vdash G_i$ reflection}
|
||||
@ -849,10 +853,10 @@
|
||||
|
||||
\[\begin{array}{rcl}
|
||||
F_iG_i(X,(B,g))
|
||||
&=& F_iW_i(G_{i-1}X,(B,g))\\
|
||||
&=& F_i(G_{i-1}X \oplus L_0^{i-1}H_\bullet(G_{i-1}X,(B,g)),\widetilde{\inj_2})\\
|
||||
&=&(F_{i-1} \times \id)\left(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,(B,g)),(A,h)\right)\\
|
||||
&=&\left(F_{i-1}(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,(B,g))),(A,h)\right)
|
||||
&=& F_iW_i(G_{i-1}X,(B,H_i\eta_{i-1}\circ g))\\
|
||||
&=& F_i\left(G_{i-1}X \tl^{i-1}K_\bullet(G_{i-1}X,(B,H_i\eta_{i-1}\circ g)),\widetilde{\inj_2}\right)\\
|
||||
&=&(F_{i-1} \times \id)\left(G_{i-1}X \tl^{i-1}K_\bullet(G_{i-1}X,(B,H_i\eta_{i-1}\circ g)),(A,h)\right)\\
|
||||
&=&\left(F_{i-1}(G_{i-1}X \tl^{i-1}K_\bullet(G_{i-1}X,(B,H_i\eta_{i-1}\circ g))),(A,h)\right)
|
||||
\end{array}\]
|
||||
|
||||
Where $(A,h)$ is the pullback defined as
|
||||
@ -863,7 +867,7 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
We can extend this pullback using the two isomorphisms given by the induction hypothesis and hypothesis H3. This pullback is over the injection morphism $\inj_2$ of the coproduct, so the new pullback object is the second component of the $\bullet \oplus B$ i.e. $B$.
|
||||
We can extend this pullback with the two isomorphisms $\en_0^i$ and $H_iF_{i-1}(\inj_\tl^{i-1})$ in another pullback. This new pullback is over the injection morphism $\inj_2$ of the coproduct of $\TSet$, so the new pullback object is the second component of the $\bullet \oplus B$ i.e. $B$.
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@ -1,5 +1,4 @@
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||||
% Loading packages
|
||||
\usepackage[utf8]{inputenc}
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\usepackage{ae}
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\usepackage[T1]{fontenc}
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\usepackage[USenglish]{babel}
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@ -111,7 +110,6 @@
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\newcommand\FamSet{{\ensuremath{\operatorname{\mathcal{F}am\mathcal{S}et}}}}
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\newcommand\Hom{{\ensuremath{\operatorname{\mathcal{H}om}}}}
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\newcommand\this{{\ensuremath{\operatorname{\texttt{this}}}}}
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\newcommand\Hbar{{\ensuremath{K}}}
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\newcommand\one{{\ensuremath{\mathbf{1}}}}
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\newcommand\dash{{\;\textrm{---}\;}}
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\renewcommand\enquote[1]{``#1''}
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@ -44,7 +44,7 @@
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||||
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||||
\subsection*{La contribution proposée}
|
||||
|
||||
Durant ce stage, j'ai apporté une formalisation sémantique de la transformation. J'ai décrit formellement la transformation de modèles du GAT original en modèle du GAT transformé, mais également la transformation réciproque. J'ai également prouvé certaines propriétés de cette transformation.
|
||||
Durant ce stage, j'ai recherché à formaliser sémantiquement cette transformation. J'ai commencé par l'appliquer à des exemples, trouver des propriétés sur ces exemple, pour ensuite formaliser dans la théorie des catégorie la transformation, et prouver formellement les propriétés que j'avais remarquées sur les exemples.
|
||||
|
||||
\question{Qu'avez vous proposé comme solution à cette question ?
|
||||
Attention, pas de technique, seulement les grandes idées !
|
||||
@ -52,9 +52,9 @@
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||||
|
||||
\subsection*{Les arguments en faveur de sa validité}
|
||||
|
||||
La preuve a été faite avec la sémantique en essayant de généraliser les objets utilisés au maximum.
|
||||
La preuve a été faite sémantiquement en essayant à chaque étape de généraliser les objets utilisés au maximum. Les constructions des objets sont basées sur des papiers déjà publiés, qui sont cités dans ce rapport.
|
||||
|
||||
Cette preuve valide la conjecture établie par Sestini dans sa thèse.
|
||||
La construction proposées et les propriétés établies prouvent également la conjecture établie par Philippo Sestini dans sa thèse.
|
||||
|
||||
\question{Qu'est-ce qui montre que cette solution est une bonne solution ?}
|
||||
\question{Des expériences, des corollaires ?}
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user