Replaced a lot of oplus to tl
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@ -118,7 +118,7 @@
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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 can add constructors on a later step, as described in \autoref{sec:constructors2trans}.
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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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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 \inlinetodo{MISSING REF}.
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@ -128,46 +128,36 @@
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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 $R_{i-1}^i : \BB_i \to \BB_{i-1}$ and $R_0^i : \BB_i \to \BB_0$, taking track of the fixed sort specification of the transformed GAT. \inlinetodo{Mal Dit, est-ce nécessaire ?}
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\item An operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ along with morphisms $\inj_1^i : X \to X \tl Y$
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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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\begin{center}
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% YADE DIAGRAM TlUniversal.json
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% GENERATED LATEX
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\input{graphs/TlUniversal.tex}
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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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\todo{Faut indiquer $\oplus = \tl^0$ à un moment :/}
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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 $F_i\inj_1$ is an isomorphism
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\item A proof that
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\end{itemize}
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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}.
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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.
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The overall recursive construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
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Here is a figure that describes the recursive construction of some of the above objects
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\begin{center}
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% YADE DIAGRAM G1.json
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% GENERATED LATEX
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\input{graphs/G1.tex}
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% END OF GENERATED LATEX
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\end{center}
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\subsection{Preliminaries}
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\paragraph{Category of models of the two-sort sort specification}
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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 a the category of models of the two-sort specification the category $\TSet$ of presheaves over the category with one arrow.
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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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\begin{center}
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% YADE DIAGRAM G0.json
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% GENERATED LATEX
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\input{graphs/G0.tex}
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% END OF GENERATED LATEX
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\end{center}
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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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\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{Grothendieck Construction}
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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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\begin{itemize}
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\item $X$ an object of $\mathcal{C}$
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@ -193,6 +183,29 @@
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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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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.
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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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\begin{center}
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% YADE DIAGRAM G0.json
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% GENERATED LATEX
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\input{graphs/G0.tex}
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% END OF GENERATED LATEX
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\end{center}
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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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\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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\todo{Rewrite this part explaining the correspondance}
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\paragraph{$\Hbar$ functor}
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Where $\Hbar_A$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ defined as
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@ -204,81 +217,79 @@
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\begin{remark}
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This functor can be constructed using the formal construction of the Grothendieck construction as a pullback in the category of categories $\Cat$
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\end{remark}
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\subsection{Initialization}
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In the first step, corresponding to an empty sort specification, the objects are defined as follows:
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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 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)
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\item $\tl^0 : \BB_0 \times \BB_0 \to \BB_0$ is the coproduct 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. \inlinetodo{Est-ce que ça pose problème de parler de \emph{the} coproduct ?}
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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 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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Therefore, we have that $F_0 \vdash G_0$ \inlinetodo{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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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}.
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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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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.
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The overall recursive construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
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\begin{center}
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% YADE DIAGRAM G1.json
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% GENERATED LATEX
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\input{graphs/G1.tex}
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% END OF GENERATED LATEX
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\end{center}
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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 $\TSet$.
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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 $L_{i-1}^i$.
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As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will use the two following adjunctions
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\[\begin{array}{c}
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R_0^i = R_{0}^{i-1} \circ R_{i-1}^{i} = R_{0}^{1} \circ ... \circ R_{i-1}^{i}\\
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L_0^i = L_{i-1}^{i} \circ L_{0}^{i-1} = L_{i-1}^{i} \circ ... \circ L_{0}^{1}
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\end{array}\]
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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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\begin{remark}
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There is also an adjunction chain between $\CC_0$,$\dots$,$\CC_{i-1}$,$\CC_i$, but we don't use it in the proof.
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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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\end{remark}
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\subsubsection{Constructing $\CC_i$}
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We construct the category $\CC_i$ as the following pair:
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We construct the category $\CC_i$ as the following Grothendieck pair:
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\[
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\CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/H_i(X)\right) = (X : \CC_{i-1}) \times \left(\Set^{H_i(X)}\right)
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\]
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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.
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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{$H_i$ functors for our Type Theory example}
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Let us give an example of those $H_i$ objects for our type theory example. We begin with
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Let us now give an example of those $H_i$ objects for our type theory example.
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We begin with the following functor, corresponding to the sort declaration $\boxed{\Con : \Set}$
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\[
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H_1(\star) = 1 \in \operatorname{Obj}(\Set)
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H_1(\star) = 1 \in \operatorname{Obj}(\Set)
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\]
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which corresponds to the fact that $\Con$ takes no parameter.
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Therefore $\CC_1 = 1 \times \Set^1 = \Set$, and the set of a model corresponds to \enquote{the set of contexts}.
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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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Then, we take the functor $H_2(X_\Con) = X_\Con$ (this means, that types need \emph{one} context to be built).
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Then, we take the functor $H_2(X_\Con) = X_\Con$, corresponding to the sort declaration $\boxed{\Ty : (\Gamma : \Con) \to \Set}$ (this means, that types need \emph{one} context to be built).
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Therefore $\CC_2 = (X:\Set) \times \Set^X \cong \FamSet$, families of sets.
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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.
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Finally, we take the functor $H_3(X_\Con,X_\Ty) = \sum_{\Gamma : X_\Con}X_\Ty(\Gamma)$ (this means that terms need \emph{one} context, and \emph{one} type of that context).
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Finally, we take the functor $H_3(X_\Con,X_\Ty) = \sum_{\Gamma : X_\Con}X_\Ty(\Gamma)$, that corresponds to the sort declaration $\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}$ (this means that terms need \emph{one} context, and \emph{one} type of that context).
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The final category $\CC_3$ is composed of triples $(X_\Con: \Set, X_\Ty : X_\Con \to \Set, X_\Tm : (\Delta: X_\Con) \to X_\Ty(\Delta) \to \Set)$
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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 given in \autoref{sec:CtoSSetFiore}.
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\end{remark}
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An object of that final category $\CC_3$ is a triple $(X_\Con: \Set, X_\Ty : X_\Con \to \Set, X_\Tm : (\Delta: X_\Con) \to X_\Ty(\Delta) \to \Set)$
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\subsubsection{Constructing $\BB_i$}
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\paragraph{The category} We construct the category $\BB_i$ as follows.
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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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\begin{itemize}
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\item an object $X$ of $\BB_{i-1}$
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\item a \enquote{sort constructor} $\Cstr_i$ as a function $H_i(F_{i-1}X) \to (R_0^{i-1}X)_\UU$
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\item a \enquote{sort constructor} $\Cstr$ as a function $H_iF_{i-1}X \to (R_0^{i-1}X)_\UU$
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\newline
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where $H_i$ is the functor $\CC_{i-1} \to \Set$ that describe the sort constructor being processed, and $F_{i-1}$ is the right part of the adjunction $\BB_{i-1} \to \CC_{i-1}$ that we are defining recursively at the same time.
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This $H_i$ functor will be so that $H_i \circ F_{i-1} \circ \inj_1^{i-1}$ is an isomorphism. \inlinetodo{Pas déclaré ici, c'est grâve ?}
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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 are defining recursively at the same time.
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\end{itemize}
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If we have an object $X$ of $\BB_i$, the first component is denoted as $R_{i-1}^iX$, and the second component is denoted as $\Cstr^X : R_{i-1}^i \to (R_0^iX)_\UU$.
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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.
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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.
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\begin{center}
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% YADE DIAGRAM D1.json
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@ -287,62 +298,15 @@
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% END OF GENERATED LATEX
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\end{center}
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For a morphism $f : X \to X'$ of $\BB_i$, we denote as $R_{i-1}^if$ the underlying morphism $R_{i-1}^i X \to R_{i-1}^i X'$.
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Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
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\paragraph{The adjunction}
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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 adjoint} of another functor we call $L_{i-1}^i$.
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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 $R_i^j \vdash L_i^j$.
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\paragraph{The coproduct}
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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 $\{f;g\} \circ \inj^i_2 = g$.
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\begin{remark}
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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 $inj_1 : G_{i-1}O_i \to G_{i-1}O_i \oplus L_0^{i-1} y\UU$. \inlinetodo{Ça ne marche plus du coup :/}
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\end{remark}
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\subsection{Induction Hypotheses}
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In order to build and prove the adjunction, we will state some recurrence invariants that we will prove after having built objects.
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\begin{property}[H1]
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The canonical morphism
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\[
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\simpleArrow{R_{i-1}^i X \oplus L_0^{i-1} Y}{\left\{R_{i-1}^i \inj_1^i ; R_{i-1}^i \inj_2^i \circ \eta_{i-1}^i\right\}}{R_{i-1}^i(X \oplus_i L_0^i Y)}
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\]
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that we will denote as $\en_{i-1}^i$ is an isomorphism.
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Its recursive version is the following isomorphism, denoted as $\en_0^i$
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\[
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\simpleArrow{ R_{0}^i X \oplus_0 Y}{\left\{R_0^i \inj_1^i ; R_0^i \inj_2^i \circ \eta_0^i\right\}}{R_0^i(X \oplus_i L_0^i Y)}
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\]
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\end{property}
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\begin{property}[H3]
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\[
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\simpleArrow{F_i X}{F(\inj_1^i)}{F_i(X \oplus L_0^i Y)}
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\]
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is an isomorphism.
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We will often this equality along with the $F_i \vdash G_i$ adjunction (for an object $O$ in $\CC_i$)
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\[
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\Hom(G_i O, X) \cong \Hom(O, F_i X) \cong \Hom(O, F_i(X \oplus L_0^i Y)) \cong \Hom(G_i O, X \oplus L_0^i Y)
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\]
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This new isomorphism is the following:
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\[
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\simpleArrow{\Hom(G_i O, X)}{(inj_1^i \circ \dash)}{\Hom(G_i O,X \oplus L_0^i Y)}
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\]
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\todo{Du coup techniquement, c'est une propriété de $H_i$. Faut réussir que c'est \emph{parce que} $H_i$ est représentable que l'on peut déduire H3' de H3.}
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\end{property}
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$R_{i-1}^i$ acts on objects and morphisms, and therefore creates a functor $\BB_i \to \BB_{i-1}$.
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\subsection{Constructing the functors}
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In order to use all the power of the recurrence, we will build the $F_i \vdash G_i$ adjunction using the $F_{i-1} \vdash G_{i-1}$ adjunction, following the diagram below.
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Build the $F_i \vdash G_i$ adjunction using the two functors of the $F_{i-1} \vdash G_{i-1}$ adjunction, following the diagram below.
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\begin{center}
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% YADE DIAGRAM G2.json
|
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@ -351,34 +315,39 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
\todo{$G_{i-1} \times \id$ et son compère, c'est bien legit ?}
|
||||
|
||||
The first part $G_{i-1} \times \id \dashv F_{i-1} \times \id$ is proven and defined as an adjunction from the previous step of the recurrence.
|
||||
Where
|
||||
\[
|
||||
(F_{i-1} \times \id)(X,(Y,y)) = (F_{i-1}X,(Y,y))
|
||||
\]
|
||||
and
|
||||
\[
|
||||
(G_{i-1} \times \id)(X,(Y,y)) = (G_{i-1}X,(Y,H_i\eta_{i-1} \circ y))
|
||||
\]
|
||||
\todo{Définir $\eta_{i-1}$ ici !}
|
||||
\todo{Plus de précision sur ces constructions ?}
|
||||
\todo{Expliquer pourquoi $(F_{i-1} \times \id) \vdash (G_{i-1} \times \id)$ est une adjonction}
|
||||
|
||||
|
||||
\subsubsection{W definition}
|
||||
|
||||
We define a functor $W : \left(X : \BB_{i-1}\right) \times \Set/H_i(F_{i-1}X) \to \BB_{i}$
|
||||
We define a functor $W : \left(X : \BB_{i-1}\right) \times \Set/H_iF_{i-1}X \to \BB_{i}$
|
||||
|
||||
The action on objects is as follows:
|
||||
\[
|
||||
W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{H_iF_{i-1}}(X,Y), \widetilde{\inj_2} \right)
|
||||
W(X,Y) := \left(X \tl^{i-1} K_{H_iF_{i-1}}(X,Y), \widetilde{\inj_2} \right)
|
||||
\]
|
||||
|
||||
With $\widetilde{\inj_2}$ being defined by \inlinetodo{Changer les noms des hypothèses H3' et H1r}
|
||||
\[
|
||||
\begin{array}{lcl}
|
||||
H_iF_{i-1}(X \oplus L_0^{i-1} \Hbar_\bullet(X,Y)))
|
||||
& \to^{\text{H3'}} & H_i(F_{i-1}X)\\
|
||||
& = & \left(\Hbar_{H_iF_{i-1}}(X,Y)\right)_\UU \\
|
||||
& \to^{\inj_2^0} & \left(R_0^{i-1}X \oplus \Hbar_\bullet(X,Y)\right)_\UU \\
|
||||
& \to^{\text{H1r}} & \left(R_0^{i-1}(X \oplus L_0^{i-1}\Hbar_\bullet(X,Y))\right)_\UU
|
||||
\end{array}
|
||||
\]
|
||||
With $\widetilde{\inj_2}$ being defined by the composition
|
||||
\begin{center}
|
||||
% YADE DIAGRAM Wdef.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/Wdef.tex}
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
The action on a morphism $(g,h)$ from $(X,Y)$ to $(X',Y')$ is the following:
|
||||
The action on a morphism $(g,h)$ from $(X,(Y,y))$ to $(X',(Y',y'))$ is the following:
|
||||
\[
|
||||
W(g,h) := \left(g \oplus L_0^{i-1} \Hbar_{H_iF_{i-1}}(g,h)\right)
|
||||
W(g,h) := \left(g \tl^{i-1} K_{H_iF_{i-1}}(g,h)\right)
|
||||
\]
|
||||
|
||||
It is indeed a morphism from $\BB_{i}$ as it makes the following diagram commute.
|
||||
@ -396,7 +365,7 @@
|
||||
|
||||
The action on objects is
|
||||
\[
|
||||
E(X) = (R_{i-1}^i X, (A,h))
|
||||
E(X) = (R_{i-1}^i X, (A,h))
|
||||
\]
|
||||
Where $(A,h)$ is defined as the following pullback:
|
||||
|
||||
@ -433,7 +402,7 @@
|
||||
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 $R_{i-1}^i f \circ inj_1^{i-1} : X \to R_{i-1}^i Z$, with $R_{i-1}^i f$ being a morphism of $\BB_{i-1}$ from $R_{i-1}^i(W(X,Y)) = X \oplus_{i-1} L_0^{i-1}$ to $R_{i-1}^i Z$.
|
||||
The first component of $\phi_{XYZ}(f)$ is $R_{i-1}^i f \circ inj_\tl^{i-1} : X \to R_{i-1}^i Z$, with $R_{i-1}^i f$ being a morphism of $\BB_{i-1}$ from $R_{i-1}^i(W(X,Y)) = X \tl^{i-1}K_{H_iF_{i-1}}(X,Y)$ to $R_{i-1}^i Z$.
|
||||
|
||||
The second component is defined through the universal property of the pullback defined by $E(Z)$ according to the following diagram:
|
||||
|
||||
@ -448,7 +417,8 @@
|
||||
|
||||
Now, we take $(g,h)$ a morphism from $(X,Y)$ to $E(Z)$.
|
||||
|
||||
We define $\phi^{-1}_{XYZ}(g,h)$ as a morphism of $\BB_i$ from $W(X,Y)$ to $Z$, i.e. a morphism of $\BB_{i-1}$ from $X \oplus_{i-1} L_0^{i-1} \Hbar (X,Y)$ to $R_0^{i-1}(Z)$ that makes a certain diagram commute:
|
||||
We define $\phi^{-1}_{XYZ}(g,h)$ as a morphism of $\BB_i$ from $W(X,Y)$ to $Z$, i.e. 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 is built using the universal property of $\tl^{i-1}$
|
||||
\todo{Describe the initial property}
|
||||
\[
|
||||
\phi^{-1}_{XYZ}(g,h) := \left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\}
|
||||
\]
|
||||
@ -465,7 +435,7 @@
|
||||
This is indeed a morphism of $\BB_i$ from $W(X,Y)$ to $Z$ as it makes the following diagram commute
|
||||
|
||||
\begin{center}
|
||||
% YADE DIAGRAM D8.json
|
||||
% YADE DIAGRAM D8.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/D8.tex}
|
||||
% END OF GENERATED LATEX
|
||||
@ -480,30 +450,29 @@
|
||||
|
||||
The proof is given in \autoref{apx:FG-refl}.
|
||||
|
||||
\subsection{Proof of the hypotheses}
|
||||
\subsubsection{Proof of H1}
|
||||
\subsection{Other objects}
|
||||
\subsubsection{Constructing $\tl^i$}
|
||||
|
||||
\todo{Relire + réeexpliquer pourquoi ça prouve}
|
||||
\label{sec:coproductConstr}
|
||||
We will define the sums of the form $X \oplus_i L_0^i Y$ in $\BB_i$.
|
||||
|
||||
\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:
|
||||
\[
|
||||
X \oplus_i L_0^i Y := \left(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y, (R_0^{i-1} \inj_1^{i-1})_\UU \circ \Cstr_i^X \circ (H_iF_{i-1}\inj_1^{i-1})^{-1}\right)
|
||||
X \tl_i Y := \left(R_{i-1}^i X \tl^{i-1} Y, (R_0^{i-1} \inj_\tl^{i-1})_\UU \circ \Cstr_i^X \circ (H_iF_{i-1}\inj_\tl^{i-1})^{-1}\right)
|
||||
\]
|
||||
|
||||
Here, $(\inj_1^{i-1} \circ \dash)^{-1}$ is the inverse of the isomorphism of hypothesis H3', and
|
||||
|
||||
|
||||
The constructor goes as follows:
|
||||
\[
|
||||
H_iF_{i-1}(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y) \equiv
|
||||
H_iF_{i-1}(R_{i-1}^i X) \to
|
||||
(R_0^{i-1} X)_\UU \to
|
||||
(R_0^i (R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y))_\UU
|
||||
\]
|
||||
|
||||
The first injector $X \to X \oplus_i L_0^i Y$ is defined as follows:
|
||||
\begin{center}
|
||||
% YADE DIAGRAM TlConstructor.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/TlConstructor.tex}
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
The injector $\inj_\tl^i : X \to X \tl^i Y$ is defined as follows:
|
||||
\[
|
||||
\inj_1^i := \inj_1^{i-1} : R_{i-1}^i X \to R_{i-1}^i (X \oplus_i L_0^i Y) = R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y
|
||||
\inj_\tl^i := \inj_\tl^{i-1} : R_{i-1}^i X \to R_{i-1}^i (X \tl^i Y) = R_{i-1}^i X \tl^{i-1} Y
|
||||
\]
|
||||
|
||||
It is a morphism of $\BB_i$ as it makes the following diagram commute:
|
||||
@ -515,20 +484,18 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
\paragraph{Universal Property}
|
||||
We will now show that that the universal property stated holds.
|
||||
To that extent, we take two objects $X$ and $Z$ in $\BB_i$, $Y$ in $\BB_0$, a morphism $g : X \to Z$ in $\BB_i$ and a morphism $h : Y \to R_0^iZ$ in $\BB_0$. We want to build a morphism $\{g,h\}$ of $\BB_i$ such that the following two diagrams commute.
|
||||
|
||||
The second injector is defined as follows:
|
||||
\[
|
||||
\inj_2^i := (\varepsilon_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
|
||||
\]
|
||||
\begin{center}
|
||||
% YADE DIAGRAM TlUniversal.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/TlUniversal.tex}
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
Where $\varepsilon_i$ is the counit of the adjunction $R_{i-1}^i \vdash L_{i-1}^i$, going from $L_{i-1}^i R_{i-1}^i X$ to $X$.
|
||||
|
||||
This goes from $L_0^i Y = L_{i-1}^i L_0^{i-1} Y$ to $L_{i-1}^i(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y)$, which is equivalent to $L_{i-1}^i R_{i-1}^i X \oplus_i L_0^i Y$ as $L_{i-1}^i$ is a left-adjoint functor and therefore it preserves colimits; then goes to $X \oplus_i L_0^i Y$.
|
||||
|
||||
We will now show that this definition is actually a definition of the coproduct in $\BB_i$.
|
||||
To that extent, we take two objects $X$ and $Z$ in $\BB_i$, $Y$ in $\TSet$ and two morphisms of $\BB_i$ $\varphi_1 : X \to Z$ and $\varphi_2 : L_0^i Y \to Z$.
|
||||
|
||||
We define $\{\varphi_1 ; \varphi_2\}_{i } = \{R_{i-1}^i \varphi_1 ; R_{i-1}^i \varphi_2 \circ \eta^i_{L_0^{i-1} Y}\}_{i-1}$ a morphism of $\BB_i$ as it makes the following diagram commute:
|
||||
We define $\{g ; h\}_i = \{R_{i-1}^i g ; h\}_{i-1}$ a morphism of $\BB_i$ as it makes the following diagram commute:
|
||||
|
||||
\begin{center}
|
||||
% YADE DIAGRAM D5.json
|
||||
@ -537,13 +504,11 @@
|
||||
% 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.
|
||||
|
||||
\todo{Justifier $R_{i-1}^i(\varepsilon_i \oplus_i \id_{L_0^i Y}) = R_{i-1}^i \varepsilon_i \oplus_{i-1} \id_{L_0^{i-1} Y}$ (with H1 ?)}
|
||||
|
||||
\subsubsection{Proof of H3}
|
||||
|
||||
We need to prove that, for any objects $(X,\Cstr)$ in $\BB_i$ and $Y$ in $\TSet$, that the morphism
|
||||
$F_i(\inj_1^i) : F_i(X,\Cstr) \to F_i((X,\Cstr) \oplus L_0^i Y)$ is an isomorphism.
|
||||
\paragraph{Composition with $F_i$}
|
||||
We finally need to prove, for any objects $X$ in $\BB_i$ and $Y$ in $\TSet$, that the morphism
|
||||
$F_i(\inj_\tl^i) : F_iX \to F_i(X \tl^i Y)$ is an isomorphism.
|
||||
|
||||
We know from \autoref{sec:coproductConstr} that $\inj_1^i := \inj_1^{i-1}$ as a morphism of $\BB_{i}$ is a morphism $\BB_{i-1}$ that verifies some equalities.
|
||||
|
||||
@ -697,6 +662,9 @@
|
||||
This equality allows us to construct the $\Gamma_i$ functors from the final $S$ category.
|
||||
\end{remark}
|
||||
|
||||
\subsection{Consructing the $H_i$ functors from the syntax}
|
||||
\label{sec:HiFromSyntax}
|
||||
|
||||
\section{Summary}
|
||||
|
||||
\lipsum[2-3]
|
||||
@ -906,6 +874,28 @@
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1 +1 @@
|
||||
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|
||||
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|
||||
@ -1 +1 @@
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Reference in New Issue
Block a user