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@ -89,7 +89,7 @@
$\dots$
\end{tabular}
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.
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}:
\begin{quote}
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.
@ -234,14 +234,14 @@
\paragraph{$K$ functor}
We define a helper functor $K : (X:C) \times (\Set/A(X)) \to \TSet$ defined as follows
We define a helper functor $K_A : (X:C) \times (\Set/A(X)) \to \TSet$ defined as follows
\[
K_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)}
K_A(X,(Y,f)) = \TSetObject{Y}{f}{A(X)}
\]
The morphisms are translated as-is.
\begin{remark}
This functor can be constructed using the formal construction of the Grothendieck construction as a pullback in the category of categories $\Cat$
This functor can be constructed using the formal construction of the Grothendieck construction of $\TSet \cong (X: \Set) \times (\Set/X)$ as a pullback in the category of categories $\Cat$
\end{remark}
\subsection{Initialization}
@ -283,6 +283,8 @@
\]
and where $H_i$ is the specific functor described above.
This way of constructing the category has formerly been described by Altenkirsch and al. \cite{Altenkirch2018}, with the only difference of the equivalence $\Set/H_i(X) \simeq \Set^{H_i(X)}$.
\paragraph{An example of $H_i$ functors}
Let us now give an example of those $H_i$ objects for our type theory example.
@ -305,7 +307,6 @@
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.
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.
\[
\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}
\]
@ -516,7 +517,7 @@
The proof is that statement is given in \autoref{apx:FG-refl}.
\subsection{Inductive step: $\tl^i$}
\subsection{Inductive step: Constructing $\tl^i$}
\label{sec:coproductConstr}
@ -602,58 +603,170 @@
And we know from the previous induction step that $F_{i-1}\inj_\tl^{i-1}$ is an isomorphism. Therefore, $F_i\inj_\tl^i$ is an isomorphism.
\section{Misc}
\section{Other Properties}
\subsection{Fibration of the category $\CC_i$}
In the formal proof, we defined $\CC_i$ as an inductive Grothendieck construction as in Altenkirsch and al. papers \cite{Altenkirch2018}.
\subsection{Fiore's Category - Fibration of the category of sorts}
But there is an other way \cite{Fiore2008} of building the category $\CC_i$ which is equivalent to the one presented above.
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 $S$: $\left[S,\Set\right]$.
With this other method, a sort specification is described by a countable simple direct category $S$ (i.e. a countable category where all the arrows follow an unique direction, and $\Hom$-sets are finite). The models of the sort specification are therefore the presheaves over the category $S$ called $\left[S,\Set\right]$.
One can understand the correspondance between those categories and sort specifications as follows:
\begin{itemize}
\item An object of the category is a sort of the specification.
\item An arrow $x$ from an object $s$ to an object $s'$ is a parameter of the sort declaration of $s$ of the for $(x : s' \dots)$.
\item The parameter $y$ of a parameter $x$ of a sort specification (i.e. the sort declaration parameter has the form $(x: s' \dots \left[y=z\right] \dots)$) is given by $z = x \circ y$.
\end{itemize}
\paragraph{Constructing $S$ as a sequence}
In order to fall back on the definition we gave in the formal proof, we will build the final countable simple direct category $S$ as a sequence $\emptyset = S_0,S_1,\dots,S_i$ where $S_i$ is the category describing the sort specification limited to the $i$ first sort declarations.
\begin{remark}
We ignore in this definition identity arrows, and we will do so in the rest of this document. Identities are the only arrows that are not «directed» in the direct category.
Interpreting the identity arrow would mean having a parameter of type $s$ to construct the sort $s$. which loops in a self-dependency.
You can assume in the rest of the document that the formalizations \enquote{all arrows} or \enquote{the arrows} pointing to/from exclude identity arrows.
\end{remark}
The input data describing the GAT will be a sequence of finite functors $\Gamma_i : S_{i-1} \to \Set$.
\todo{Éventuellement changer tous les paramètres par la forme complète, exemple
\[
\operatorname{eq}: (\Gamma : \Con) \to (A : \Ty \left[\Gamma=\Gamma\right]) \to \Tm \left[\Gamma=\Gamma\right] \left[A=A\right] \to \Tm \left[\Gamma=\Gamma\right] \left[A=A\right] \to \Ty \left[\Gamma=\Gamma\right]
\]
C'est bien plus verbeux et en pratique pas utilisé, mais permet de mieux voir la «composition» dans la catégorie de Fiore.}
\todo{Est-ce qu'on fait une notation \enquote{arrow*} pour dire «flèche qui n'est pas l'identité» pour plus de rigueur ?}
We will then construct the category $S_i$ by adding a single object $\alpha_i$ to the category $S_{i-1}$, along with morphisms $f : \alpha_j \to \alpha_{i+1}$ for $f \in \Gamma_{i+1}(\alpha_j)$ and $j \leq i$. The morphisms follow the composition condition, describing that every pair of morphisms $f : \alpha_j \to \alpha_{i+1}$ and $g : \alpha_k \to \alpha_{i+1}$ (i.e. $f\in\Gamma_{i+1}(\alpha_k)$ and $g\in\Gamma_{i+1}(\alpha_j)$) and for every morphism of $S_{i}$ $h : \alpha_j \to \alpha_k$, we have $\Gamma_{i+1}(h)(f) \circ f = g$.
For example the category version of the specification of sorts of Type Theory given above is defined as:
Avec le formalisme de la construction de Grothendieck, on peut formaliser la construction de la nouvelle catégorie comme cela:
\[
S_i = S_{i-1} \times
\]
\begin{itemize}
\item There is three objects called $\Con$,$\Ty$, and $\Tm$.
\item The arrows are defined as
\begin{itemize}
\item There is no arrow going out of $\Con$
\item There is one arrow going out of $\Ty$: $\Gamma$ pointing to $\Con$.
\item There is two arrows going out of $\Tm$: $\Delta$ pointing to $\Con$ and $\Gamma$ pointing to $\Ty$.
\end{itemize}
\item The $\Gamma$ parameter of $\Ty$ in the parameter $A$ of $\Tm$ is $\Delta$. Therefore, we have $\Delta = A \circ \Gamma$.
\end{itemize}
The final category is simply $S = \bigcup S_i$.
The category is pictured below:
In our type theory example, the category $S$ is constructed as follows:
\begin{center}
% YADE DIAGRAM B1.json
% GENERATED LATEX
\input{graphs/B1.tex}
% END OF GENERATED LATEX
\begin{tabular}{c|c|c}
$\boxed{\Con : \Set}$ &
$\boxed{\Ty : (\Gamma : \Con) \to \Set}$ &
$\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}$ \\
$S_1 = \qquad \Con$ &
$S_2 = \qquad \Con \leftarrow \Ty$ &
$S_3 = \qquad \Con \leftarrow \Ty \leftarrow \Tm$ \\
$\left[S_1,\Set\right] \simeq \CC_1$ &
$\left[S_2,\Set\right] \simeq \CC_2$ &
$\left[S_3,\Set\right] \simeq \CC_3$
\end{tabular}
\end{center}
\begin{remark}
This definition is reversable (up to the order of the sorts), and therefore, one can define a sequence of $\Gamma_i$ functors from any countable simple direct category.
\end{remark}
\paragraph{Falling back on the formal proof}
In the formal proof, we used categories $\CC_i$ and functors $H_i : \CC_i \to \Set$. We now undertand that the categories $\CC_i$ are defined on top of the categories $S_i$ with $\CC_i := \left[S_i,\Set\right]$.
In the same way, the functors $H_i : \CC_{i-1} \to \Set$ are defined on top of the functors $\Gamma_i$ with the following relation:
\[
H_i(X) := \Hom_{\left[S_{i-1},\Set\right]}\left(\Gamma_i,X\right)
\]
\subsection{Creating semantics object from the syntax}
\label{sec:HiFromSyntax}
In this part, we will see how the functors $\Gamma_i$ given above can be constructed from the syntax of a sort specification. The previous subsection gives us a way of constructing the $H_i$ functors from the $\Gamma_i$ functors, and thus one can make the complete proof with the definition below.
\paragraph{Verbose sort specification}
We will take a verbose version of a sort specification, where when \enquote{implementing} a sort, we gave the name of the parameter and the name of the term. For example, our type theory example becomes the following verbose version:
\begin{center}
\renewcommand\arraystretch{1.5}
\begin{tabular}{l}
$\Con : \Set$\\
$\Ty : (\Gamma : \Con) \to \Set$ \\
$\Tm : (\Delta : \Con) \to (A : \Ty \left[\Gamma \mapsto \Delta\right]) \to \Set$
\end{tabular}
\end{center}
\vspace{1em}
The only sort declaration that changes is that of $\Tm$ as it is the only one for which one of its parameter is a sort with parameters.
With that verbose definition, a generic sort specification is a list of sort declarations indexed by $\mathbb{T}$, the fully ordered set of all sorts. The sort declarations are as follow, for a sort $T \in \mathbb{T}$
\[
T : \left[x_a^T : U_T(a) \left[x^{U_T(a)}_{b} \mapsto x^T_{v_T(a,b)}\right]_{b\leq I_{U_T(a)}}\right]_{a \leq I_T} \to \Set
\]
Where
\begin{itemize}
\setlength\itemsep{-.2em}
\item $I_T$ is the number of parameters of the sort $T$
\item $x^T_a$ is the $a$-th parameter of the sort $T$
\item $U_T(a)$ is the sort of the $a$-th parameter of the sort $T$ (we have $U_T(a) < T$)
\item $I_{U_T(a)}$ is the number of parameters of the sort $U_T(a)$ i.e. the number of arguments we have to give to make an object of sort $U_T(a)$
\item $x^{U_T(a)}_b$ is the $b$-th parameter of the sort $U_T(a)$
\item $v_T(a,b)$ is the index of the parameter of the sort $T$ given as the $b$-th parameter of the $a$-th parameter of the sort $T$ (we have $v_T(a,b) < a$)
\item The types of parameters have to be respected, therefore we must have the equality
\[
U_T(v_T(a,b)) = U_{U_T(a)}(b)
\]
\end{itemize}
For example, for the above sort declaration, we would have
\begin{itemize}
\setlength\itemsep{-.2em}
\item $\mathbb{T} = \left\{\Con < \Ty < \Tm\right\}$
\item $I_\Con = 0$, $I_\Ty = 1$, $I_\Tm = 2$
\item $x^\Ty_1 = "\Gamma"$, $x^\Tm_1 = "\Delta"$, $x^\Tm_2 = "A"$
\item $U_\Ty(1) = \Con$, $U_\Tm(1) = \Con$, $U_\Tm(2) = \Ty$
\item $v_\Tm(2,1) = 1$ such that $x^\Tm_{v_\Tm(2,1)} = "\Delta"$ (associated to $x^\Ty_{b=1} = "\Gamma"$)
\end{itemize}
\paragraph{Describing the category $S$}
We will then describe what is the category $S$ corresponding to a given sort specification $(\mathbb{T},I,U,v)$
Its objects are simply the sorts, i.e. the elements of $\mathbb{T}$.
Morphisms from a sort $T_x$ to a different sort $T_y$ are described by $U$ and correspond to the parameters of the sort $T_x$ that are of sort $T_y$. In other words:
\[
\Hom_{S}(T_x,T_y) = \left\{x^{T_x}_a\middle| a \leq I_{T_x} \text{ and } U_{T_x}(a) = T_y\right\}
\]
We can see that the category is direct, because of the rule $U_T(a) < T$ i.e. if $T_x < T_y$ then $\Hom_S(T_x,T_y) = \emptyset$.
For every sort $T$, we define $\Hom_S(T,T)$ to contain only the identity morphism $\id_T$, for which composition rules are already defined.
Composition is described by $v$, corresponding to\enquote{parameters of parameters}.
Let's take two morphisms $x^{T_x}_b : T_x \to T_y$ and $x^{T_y}_a : T_y \to T_z$. Their composition is defined by
\[
x^{T_y}_a \circ x^{T_x}_b = x^{T_x}_{v_{T_x}(a,b)}
\]
This composition is well-defined as
\[
U_{T_x}(v_{T_x}(a,b)) = U_{U_{T_x}(a)}(b) = U_{T_y}(b) = T_z
\]
For our type theory example, the category $S$ is as follows
\begin{center}
% YADE DIAGRAM TyThExampleSCat.json
% GENERATED LATEX
\input{graphs/TyThExampleSCat.tex}
% END OF GENERATED LATEX
\end{center}
\paragraph{Creating the functor $\Gamma_i$}
We know we can create the functors $\Gamma_i$ directly from a complete category $S$, but we will look more in detail what those functors mean related to the syntax.
If we are given the category $S_{i-1}$ with objects corresponding to the $i-1$ sorts already defined.
We want to build a finite functor $\Gamma_i : S_{i-1} \to \Set$ corresponding to the following sort declaration of a new sort $T_i$
We will build $\Gamma_i$ as follows:
\[
\Gamma_i(T_x) = \left\{x^{T_i}_a\middle| a \leq I_{T_i} \text{ and } U_{T_i}(a) = T_x\right\}
\]
\[
\Gamma_i(x_b^{T_x} : T_x \to T_y)(x^{T_i}_a \in \Gamma_i(T_x)) = x^{T_i}_{v_{T_i}(a,b)}
\]
This definition is redundant with that of the last paragraph, but it has the advantage of directly creating the functor $\Gamma_i$ that are used to create the functors $H_i$ and all the rest of the proof.
\subsection{Non-recursive definitions}
\subsubsection{Redefinition of $\BB_i$}
\subsubsection{Redefinition of $G_i$ and $F_i$}
\subsubsection{Non-direct base category}
\subsection{Infinite construction of $\BB_i$}
\[
\BB_i := \left(X : \TSet, \Cstr : (a : S_{i-1}) \to \Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,R_{a-1}^i(\this)) \to X(\UU)\right)
@ -695,46 +808,6 @@
\todo{Show that those are the same functors as those defined recursively. Prove the adjunction/reflection infinitely ?}
\subsection{Adding 2-transformation of constructors}
\label{sec:constructors2trans}
\todo{Describe the process}
\subsection{Overview}
\subsubsection{$\CC$ as presheaf category}
\label{sec:CtoSSetFiore}
We use the specification of sorts definition of Fiore \cite{Fiore2008}.
A specification of sorts is given by a sequence of functors $\Gamma_i : S_{i-1} \to \Set$. We construct the category $S_{i+1}$ by adding a single object $\alpha_{i+1}$ to the category $S_{i}$, along with morphisms $f : \alpha_j \to \alpha_{i+1}$ for $f \in \Gamma_{i+1}(\alpha_j)$ and $j \leq i$. The morphisms follow the composition condition, describing that every pair of morphisms $f : \alpha_j \to \alpha_{i+1}$ and $g : \alpha_k \to \alpha_{i+1}$ (i.e. $f\in\Gamma_{i+1}(\alpha_k)$ and $g\in\Gamma_{i+1}(\alpha_j)$) and for every morphism of $S_{i}$ $h : \alpha_j \to \alpha_k$, we have $\Gamma_{i+1}(h)(f) \circ f = g$.
We have define a sequence of direct categories $\emptyset = S_0 \subset S_1 \subset S_2 \subset \dots$ (with inclusions functors $I_i : S_{i+1} \to S_i$). We define the \enquote{final} direct category as $S = \bigcup S_i$
This definition is an isomorphism, so we can define a GAT categorically as any locally finite direct category.
Then the semantics of the GAT is described as the category of presheaves over $S$, written $[S, \Set]$.
Altenkirsch has another way of constructing the semantics of a specification of sorts \cite{Altenkirch2018}, and he also describes a way to describe constructors.
So we can construct the base category, which is that of families of sets.
In order to construct the $i$-th sort, we use a finite functor $\Gamma_i : S_{i-1} \to \Set$ describing entirely the sort declaration.
This functor is to be understood as $\Gamma_i(a)$ is the set of parameters of type $a$ for our new sort. In the above example, we would have $\Gamma_\Ty(\Con) = \{"\Gamma"\} = 1$ and $\Gamma_\Tm(\Con) = \{\Delta\}$,$\Gamma_\Tm(\Ty) = \{"A"\}$,$\Gamma_\Tm(\Gamma) = \left["A" \mapsto "\Delta"\right]$.
Then, to construct $S_i$, we add one object $i$ to $S_{i-1}$, along with morphisms $x : i \to a$ for every $x \in \Gamma_i(a)$ for every $a$ in $S_{i-1}$. We also add equalities
$s \circ x = x'$ for every $s : b \to a$ and $x \in \Gamma_i(a)$ and $x' \in \Gamma_i(b)$ where $\Gamma_i(s)(x') = x$.
\begin{remark}
We have that $\Hom_{S_i}(a,b) = \Gamma_b(a)$ or $(a/S_i)* \equiv \Gamma_a$.\inlinetodo{C'est sûr la deuxième partie ?}
This equality allows us to construct the $\Gamma_i$ functors from the final $S$ category.
\end{remark}
\subsection{Consructing the $H_i$ functors from the syntax}
\label{sec:HiFromSyntax}
\section{Summary}
\lipsum[2-3]
@ -1017,5 +1090,6 @@

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@ -0,0 +1 @@
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