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Hom(GΓ,-) ---> H

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@ -217,28 +217,29 @@
We construct the category $\CC_i$ as the following pair: We construct the category $\CC_i$ as the following pair:
\[ \[
\CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/\Hom(O_i,X)\right) \CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/H_i\right) = (X : \CC_{i-1}) \times \left(\Set^{H_i}\right)
\] \]
and where $O_i$ is a specific object of the category $\CC_{i-1}$, such that $\Hom(O_i,X)$ is the set of parameters for the construction of the new sort. and where $H_i$ is a specific representable functor $\CC_{i-1} \to \Set$, such that $H_i(X)$ is the set of parameters for the construction of the new sort.
I will give now an example of those $O_i$ objects for our type theory example. We begin with \paragraph{$H_i$ functors for Type Theory}
I will give now an example of those $H_i$ objects for our type theory example. We begin with
\[ \[
O_1 = \star \in \operatorname{Obj}(\CC_0) = \operatorname{Obj}(1) H_1(\star) = 1 \in \operatorname{Obj}(\Set)
\] \]
so $\Hom(O_1,X) = 1$, which corresponds to the fact that $\Con$ takes no parameter. which corresponds to the fact that $\Con$ takes no parameter.
Therefore $\CC_1 = 1 \times \Set^1 = \Set$ Therefore $\CC_1 = 1 \times \Set^1 = \Set$, and the set of a model corresponds to \enquote{the set of contexts}.
Then, we take the singleton object $O_2 = 1$ (this means, that types need \emph{one} context to be built), and so, for a set $X_\Con$, $\Hom(O_2,X_\Con) \cong X_\Con$, which corresponds to the fact that $\Ty$ take one $\Con$ as a parameter. Then, we take the functor $H_2(X_\Con) = X_\Con$ (this means, that types need \emph{one} context to be built).
Therefore $\CC_2 = (X:\Set) \times \Set^X \cong \FamSet$. Therefore $\CC_2 = (X:\Set) \times \Set^X \cong \FamSet$, families of sets.
Finally, we take the object $O_3 = (1, \lambda \star . 1)$ (this means that terms need \emph{one} context, and \emph{one} type of that context). With this object, for a pair $(X_\Con,X_\Ty)$ in $\CC_2$, we have $\Hom(O_3,(X_\Con,X_\Ty)) \cong \left(\Gamma: X_\Con, A: X_\Ty(\Gamma)\right)$. 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).
The final category $\CC_3$ is composed of triples $(X_\Con: \Set, X_\Ty : X_\Con \to \Set, X_\Tm : (\Delta: X_\Con) \to X_\Ty(\Delta) \to \Set)$ The final category $\CC_3$ is composed of triples $(X_\Con: \Set, X_\Ty : X_\Con \to \Set, X_\Tm : (\Delta: X_\Con) \to X_\Ty(\Delta) \to \Set)$
\begin{remark} \begin{remark}
There is a way of getting the object $O_i$ from the syntax, which is given in \autoref{sec:CtoSSetFiore}. There is a way of getting the functor $H_i$ from the syntax, which is given in \autoref{sec:CtoSSetFiore}.
\end{remark} \end{remark}
\subsubsection{Constructing $\BB_i$} \subsubsection{Constructing $\BB_i$}
@ -248,9 +249,10 @@
An object of $\BB_i$ is An object of $\BB_i$ is
\begin{itemize} \begin{itemize}
\item an object $X$ of $\BB_{i-1}$ \item an object $X$ of $\BB_{i-1}$
\item a \enquote{sort constructor} $\Cstr_i$ as a function $\Hom_{\BB_{i-1}} (G_{i-1}O_i,X) \to (R_0^{i-1}X)_\UU$ \item a \enquote{sort constructor} $\Cstr_i$ as a function $H_i(F_{i-1}X) \to (R_0^{i-1}X)_\UU$
\newline \newline
where $O_i$ is the object of $\CC_{i-1}$ that describe the sort constructor being processed, and $G_{i-1}$ is the left part of the adjunction $\CC_{i-1} \to \BB_{i-1}$ that we are defining recursively at the same time. 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.
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 ?}
\end{itemize} \end{itemize}
@ -268,13 +270,13 @@
\paragraph{The adjunction} \paragraph{The adjunction}
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 $L_{i-1}^i$. 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 $L_{i-1}^i$.
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 $R_i^j \vdash L_i^j$. 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 $R_i^j \vdash L_i^j$.
\paragraph{The coproduct} \paragraph{The coproduct}
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 $\{f;g\} \circ \inj^i_2 = g$. 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 $\{f;g\} \circ \inj^i_2 = g$.
\begin{remark} \begin{remark}
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 $inj_1 : G_{i-1}O_i \to G_{i-1}O_i \oplus L_0^{i-1} y\UU$. 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 :/}
\end{remark} \end{remark}
@ -312,6 +314,8 @@
\simpleArrow{\Hom(G_i O, X)}{(inj_1^i \circ \dash)}{\Hom(G_i O,X \oplus L_0^i Y)} \simpleArrow{\Hom(G_i O, X)}{(inj_1^i \circ \dash)}{\Hom(G_i O,X \oplus L_0^i Y)}
\] \]
\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.}
\end{property} \end{property}
\subsection{Constructing the functors} \subsection{Constructing the functors}
@ -325,23 +329,26 @@
% END OF GENERATED LATEX % END OF GENERATED LATEX
\end{center} \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. 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.
\subsubsection{W definition} \subsubsection{W definition}
We define a functor $W : \left(X : \BB_{i-1}\right) \times \Set/\Hom_{\BB_{i-1}}(G_{i-1}O_i,X) \to \BB_{i}$ We define a functor $W : \left(X : \BB_{i-1}\right) \times \Set/H_i(F_{i-1}X) \to \BB_{i}$
The action on objects is as follows: The action on objects is as follows:
\[ \[
W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}O_i,\dash)}(X,Y), \widetilde{\inj_2} \right) W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{H_iF_{i-1}}(X,Y), \widetilde{\inj_2} \right)
\] \]
With $\widetilde{\inj_2}$ being defined by \inlinetodo{Changer les noms des hypothèses H3' et H1r} With $\widetilde{\inj_2}$ being defined by \inlinetodo{Changer les noms des hypothèses H3' et H1r}
\[ \[
\begin{array}{lcl} \begin{array}{lcl}
\Hom(G_{i-1}O_i,X \oplus L_0^{i-1} \Hbar_\bullet(X,Y)) & \to^{\text{H3'}} & \Hom(G_{i-1}O_i,X)\\ H_iF_{i-1}(X \oplus L_0^{i-1} \Hbar_\bullet(X,Y)))
& = & \Hbar_\bullet(X,Y)_\UU \\ & \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^{\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 & \to^{\text{H1r}} & \left(R_0^{i-1}(X \oplus L_0^{i-1}\Hbar_\bullet(X,Y))\right)_\UU
\end{array} \end{array}
@ -349,7 +356,7 @@
The action on a morphism $(g,h)$ from $(X,Y)$ to $(X',Y')$ is the following: The action on a morphism $(g,h)$ from $(X,Y)$ to $(X',Y')$ is the following:
\[ \[
W(g,h) := \left(g \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}O_i,\dash)}(g,h)\right) W(g,h) := \left(g \oplus L_0^{i-1} \Hbar_{H_iF_{i-1}}(g,h)\right)
\] \]
It is indeed a morphism from $\BB_{i}$ as it makes the following diagram commute. It is indeed a morphism from $\BB_{i}$ as it makes the following diagram commute.
@ -363,7 +370,7 @@
\subsubsection{E definition} \subsubsection{E definition}
We define $E : \BB_{i} \to \left(X : \BB_{i-1}\right) \times (\Set/\Hom_{\BB_{i-1}}(G_{i-1}O_i,X))$ We define $E : \BB_{i} \to \left(X : \BB_{i-1}\right) \times (\Set/H_iF_{i-1}X)$
The action on objects is The action on objects is
\[ \[
@ -391,7 +398,7 @@
We prove that $(W,E)$ make an adjunction showing that there is a natural isomorphism between $\Hom$ sets in both categories. We prove that $(W,E)$ make an adjunction showing that there is a natural isomorphism between $\Hom$ sets in both categories.
We want to construct for each $(X,Y)$ in $\displaystyle\prod_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}O_i,X))$ and $Z$ in $\BB_i$, an isomorphism $\phi_{XYZ}$. We want to construct for each $(X,Y)$ in $(X : \BB_{i-1}) \times (\Set/H_iF_{i-1}X)$ and $Z$ in $\BB_i$, an isomorphism $\phi_{XYZ}$.
\[ \[
\phi_{XYZ} : \Hom(W(X,Y),Z) \to \Hom((X,Y),E(Z)) \phi_{XYZ} : \Hom(W(X,Y),Z) \to \Hom((X,Y),E(Z))
@ -466,7 +473,10 @@
The constructor goes as follows: The constructor goes as follows:
\[ \[
\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y) \equiv \Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,R_{i-1}^i X) \to (R_0^{i-1} X)_\UU \to (R_0^i (R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y))_\UU H_iF_{i-1}(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y) \equiv
H_iF_{i-1}(R_{i-1}^i X) \to
(R_0^{i-1} X)_\UU \to
(R_0^i (R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y))_\UU
\] \]
The first injector $X \to X \oplus_i L_0^i Y$ is defined as follows: The first injector $X \to X \oplus_i L_0^i Y$ is defined as follows:
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