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@ -23,10 +23,10 @@
\newpage
\section{Sort specification}
\section{Introduction}
A Generalized Algebraic Theory (or GAT), first introduced by Cartmell \cite{CartmellGATs}, is a syntactic specification of an algebraic structure. From a GAT, one can define a category of models describing the models of the algebraic structure.
A GAT starts with a sort specification i.e. a list of sort declarations, eventually followed by a list of constructors.
A GAT typically starts with a sort specification i.e. a list of sort declarations, eventually followed by a list of constructors.
In this document, we will not ask ourselves about the specific syntax of GATs, a \enquote{vague} definition is enough.
\paragraph{Sort specification}
@ -46,9 +46,9 @@
A model of this category is a triple
\begin{itemize}
\item A set $X_\Con : \Set$
\item A family of sets $\left(X_\Ty\left(\Gamma\right)\right)_{\Gamma \in _\Con}$
\item A family of sets $\left(X_\Tm\left(\Delta,A\right)\right)_{\Delta\in X_\Con,\: A \in X_\Ty\left(\Delta\right)}$
\item A set $X_\Con$ of contexts
\item A family of sets $\left(X_\Ty\left(\Gamma\right)\right)_{\Gamma \in _\Con}$ of types, indexed by contexts
\item A family of sets $\left(X_\Tm\left(\Delta,A\right)\right)_{\Delta\in X_\Con,\: A \in X_\Ty\left(\Delta\right)}$ of terms, indexed by contexts and types.
\end{itemize}
\paragraph{Constructor specification}
@ -58,87 +58,62 @@
\vspace{1em}
\renewcommand\arraystretch{1.5}
\begin{tabular}{p{.37\textwidth}|p{.6\textwidth}}
$\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ & In any context $\Gamma$, a type of $\Ty\;\Gamma$ called unit.\\
$\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ & In any context $\Gamma$, a type of $\Ty\;\Gamma$ called unit.\\
$\operatorname{eq}: (\Gamma : \Con) \to (A : \Ty\;\Gamma) \to$
$\qquad\Tm\;\Gamma A \to \Tm\;\Gamma A \to \Ty\;\Gamma$ & In any context $\Gamma$ and type $A$ in this context, for every terms $t$,$u$ of the type $A$, we have a type $\operatorname{eq} \Gamma A t u$ describing the equality of the terms.
$\operatorname{tt}: (\Gamma : \Con) \to \Tm\;\Gamma\;(\operatorname{unit}\;\Gamma)$ & In any context $\Gamma$, we have a term whose type is the $\operatorname{unit}$ of this context ($\operatorname{unit}\;\Gamma$).
\end{tabular}
\vspace{1em}
This adds to the previous model two functions that \enquote{points} one element of the sets
\begin{itemize}
\item For each $\Gamma \in X_\Con$, an element $\operatorname{unit}\;\Gamma \in X_\Ty(\Gamma)$
\item For each $\Gamma \in X_\Con$, for each $A \in X_\Ty(\Gamma)$, for each elements $u,v \in X_\Tm(\Gamma,A)$, an element $\operatorname{eq}\;\Gamma\;A\;u\;v \in X_\Ty(\Gamma)$
\item For each $\Gamma \in X_\Con$, an element $\operatorname{tt}\;\Gamma \in X_\Tm(\Gamma,\operatorname{unit}\;\Gamma)$
\end{itemize}
Sort declarations describe the sets that the model contains, whereas the constructors describe elements of these sets.
\paragraph{Two-sortification}
There is a process that allows us to transform a GAT into a GAT with only two sorts. This process is used by Philippo Sestini in his thesis \cite{SestiniPhD} refering the work of Zongpu Szumi Xie \cite{AmbrusSzumiXie2sort}:
\begin{quote}
Many instances of multi-sorted IITs [IITs are another type 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.
\end{quote}
The goal of this document is to prove semantically that this transformation makes sense. More specifically, we prove that this transformation is a left adjunct functor of a coreflection. This is enough to prove what Sestini conjectured, i.e. that the initial object in the 2-sort category creates back the initial object of the primary category \cite[5. General]{nlab:reflective_subcategory}.
We will now present this transformation. The sort specification of the transformed GAT is always the same, and contains two sort declarations (as planned):
There is a process that allows us to transform a GAT into a GAT with only two sorts.
The sort specification of the transformed GAT is always the same, and contains these two sort declarations:
\vspace{1em}
\begin{tabular}{p{0.37\textwidth}|p{0.5\textwidth}}
$\mathcal{O} : \Set$ & The set of sorts \\
$\underline{\;\bullet\;} : \mathcal{O} \to \Set$ & For every sort object $o$ in the set of sorts, a set called $\underline{o}$ of objects corresponding to the sort object.
$\UU : \Set$ & The set of sorts \\
$\El : \mathcal{O} \to \Set$ & For every sort object $o$ in the set of sorts, a set called $\El(o)$ of objects corresponding to the sort object. We will write this set $\underline{o}$ rather than $\El(o)$.
\end{tabular}
\vspace{1em}
Category of models of this two-sort specification are intuitively the category of families of set $\FamSet$, composed of pairs $\left(X_0:\Set,X_1: X_0 \to \Set\right)$.
Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply underline to every parameter. For example, the Type Theory GAT presented above becomes that which follows:
Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply $\El$ to every parameter. For example, the Type Theory GAT presented above becomes that which follows:
\begin{tabular}{p{0.4\textwidth}|p{0.5\textwidth}}
$\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\
$\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\
$\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}$ &
For each object $\Gamma$ corresponding to the sort object $\Con$, another sort object called \enquote{$\Ty\;\Gamma$} \\
$\Tm : (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}$ &
For each object $\Gamma$ corresponding to the sort object $\Con$,
and for every object $A$ corresponding to the sort object $\Ty\;\Gamma$, another sort object called \enquote{$\Tm\;\Gamma\;A$}\\
$\operatorname{unit} : (\Gamma : \underline{\Con}) \to \underline{\Ty\;\Gamma}$ &
For each object $\Gamma$ corresponding to the sort object $\Con$, an object called \enquote{$\operatorname{unit} \Gamma$} corresponding to the sort object $\Ty\;\Gamma$\\
$\operatorname{eq}: (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Gamma}) \to$ \newline
$\qquad\underline{\Tm\;\Gamma A} \to \underline{\Tm\;\Gamma A} \to \underline{\Ty\;\Gamma}$ &
and for every object $A$ corresponding to the sort object $\Ty\;\Gamma$, another sort object called "$\Tm\;\Gamma\;A$"\\
$\operatorname{unit} : (\Gamma : \underline{\Con}) \to \underline{\Ty\;\Gamma}$ &
For each object $\Gamma$ corresponding to the sort object $\Con$, an object called "$\operatorname{unit} \Gamma$" corresponding to the sort object $\Ty\;\Gamma$\\
$\operatorname{tt}: (\Gamma : \underline{\Con}) \to \underline{\Tm\;(\operatorname{unit}\;\Gamma)}$ &
$\dots$
\end{tabular}
\paragraph{$\FamSet$ as functors}
This process has been observed by Zongpu Szumi Xie \cite{AmbrusSzumiXie2sort}, and Philippo Sestini used it in his thesis \cite{SestiniPhD}:
In the rest of the document, we will denote the simple category containing two elements and one non-identity arrow between them as $\TT$. The objects and arrow of this category are pictured below.
\begin{center}
% YADE DIAGRAM G0.json
% GENERATED LATEX
\input{graphs/G0.tex}
% END OF GENERATED LATEX
\end{center}
The functors over this categories are equivalent to families of sets, using the following mapping :
\[
\begin{array}{l|l}
X_\UU = X_0 & X_0 = X_\UU \\
X_\El = \displaystyle\coprod_{A\in X_0}X_1(A) & X_1 = A \mapsto X_p^{-1}(\{A\})\\
X_p = (A,B) \mapsto A &
\end{array}
\]
Therefore the categories of sorts of the transformed GATs will be built atop of the category $\TSet$ rather than atop of the category $\FamSet$ as it makes the formal proofs more elegant.
\begin{quote}
Many instances of multi-sorted IITs [IITs are another type 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.
\end{quote}
\paragraph{Goal}
The goal of this document is to make a relation between the category of models of the GAT $\CC$ and the category of models of the two-sortified GAT $\BB$. This relation will be an adjunction $F \vdash G$ that we will prove to be a coreflection.
The goal my internship was to formally study this transformation, and to try to find a relation between the semantics of a GAT and its two-sorted version.
The category $\BB$ is built with an adjunction $R \vdash L$ to the category of models of the simple two-sort specification of sorts $\TSet$.
We managed to construct a coreflection between a category of models and the category of models of the transformed GAT.
The existence of this coreflection is enough to prove what Sestini conjectured.
We will give a formal definition of this coreflection in next section. It will consist 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.
The category $\BB$ will be built 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 constructor.
\begin{center}
% YADE DIAGRAM G1-0.json
@ -147,29 +122,104 @@
% END OF GENERATED LATEX
\end{center}
\section{Construction of the coreflection}
\subsection{Preliminaries}
\paragraph{Category of models of the two-sort sort specification}
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, in order to have more elegant constructions, we will use a the category of models of the two-sort specification the category $\TSet$ 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.
\begin{center}
% YADE DIAGRAM G0.json
% GENERATED LATEX
\input{graphs/G0.tex}
% END OF GENERATED LATEX
\end{center}
With this formalisation, a model of the two-sort GAT is a functor $X : \TSet$, such that
\begin{itemize}
\item $X_\UU$ is the set of the \enquote{sort objects}
\item For each sort object $\Gamma \in X_\UU$, the set of objects corresponding to the sort object is $X_p^-1(\{\Gamma\}) \subset X_\El$
\end{itemize}
Therefore the categories of models of the transformed GATs will be built atop of this category $\BB_0 = \TSet$.
\paragraph{Grothendieck Construction}
For a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \Cat$, the Grothendieck construction is a category whose objects are pairs of
\begin{itemize}
\item $X$ an object of $\mathcal{C}$
\item an object of $F(X)$
\end{itemize}
The morphism $(X,Y) \to (X',Y')$ is therefore a pair of a morphism $f : X \to X'$ in $\mathcal{C}$ and a morphism $g : F(f)(Y) \to Y'$ in $H(X')$.
We will denote this category $(X : \mathcal{C}) \times F(X)$ as its objects are pairs. It can some times be found written as $\int^\mathcal{C} F$ or $\sqint^\mathcal{C} F$
\paragraph{Slice category}
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
\begin{itemize}
\item $Y$ an object of $\mathcal{C}$
\item an arrow $X \to Y$ of $\mathcal{C}$
\end{itemize}
A morphism $(Y,f) \to (Y',f')$ is a morphism $g : Y \to Y'$ such that $g \circ f = f'$.
We can deduce a functor $\left(\mathcal{C}/\dash\right) : \mathcal{C} \to \Cat$ from this construction.
If the category $\mathcal{C}$ is $\Set$, we have that $\Set/X \cong \Set^X$.
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 $(X : \mathcal{C}) \times \left(\Set\middle/H(X)\right)$.
\paragraph{$\Hbar$ functor}
Where $\Hbar_A$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ defined as
\[
\Hbar_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)}
\]
The morphisms are translated as-is.
\begin{remark}
This functor can be constructed using the property of the Grothendieck construction
\end{remark}
\subsection{Constructing the categories}
We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one.
The categories $\CC_i$ are described as in Fiore's paper \cite{Fiore2008}, and the categories $\BB_i$ are constructed atop of the category $\TSet$ with a method inspired by the category of models described by Altenkirch et al. \cite{Altenkirch2018}.
We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one for each constructor. At each recursion step, we will build the categories, the adjunction, and keep some invariants that will be stated in \autoref{sec:hypotheses}.
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. $\BB_i$ will samewise be the category of models of the 2-sorted $i$ first sorts of the sort specification.
The overall recursive construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below.
\begin{center}
% YADE DIAGRAM G1.json
% GENERATED LATEX
\input{graphs/G1.tex}
% END OF GENERATED LATEX
\end{center}
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 functors $R_{i-1}^i$ are the forgetful monadic functors that forget about the $i$-th sort contsructor. They have a left adjoint denoted $L_{i-1}^i$.
As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will use the two following adjunctions
\[\begin{array}{c}
R_0^i = R_{0}^{i-1} \circ R_{i-1}^{i} = R_{0}^{1} \circ ... \circ R_{i-1}^{i}\\
L_0^i = L_{i-1}^{i} \circ L_{0}^{i-1} = L_{i-1}^{i} \circ ... \circ L_{0}^{1}
\end{array}\]
\begin{remark}
There is also an adjunction chain between $\CC_0$,$\dots$,$\CC_{i-1}$,$\CC_i$, but we don't use it in the proof.
\end{remark}
\subsubsection{Constructing $\CC_i$}
We construct the category $\CC_i$ as the following pair:
\[
\CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/\Hom(O_i,X)\right)
\]
where for any category $\mathcal{C}$ and $X$ an object of $\mathcal{C}$, $(\mathcal{C}/X)$ it the slice category (or over category) of arrows pointing out of $X$ (its objects $(Y,f)$ are arrows $f : X \to Y$ and its morphisms are morphisms creating commutative triangles).\inlinetodo{Assez clair ?} \inlinetodo{On ne voit pas que $(\Set/A(X)) \cong \Set^{A(X)}$}
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.
\todo{Comment indiquer que la paire est dépendante ?}
I will give now an example of those $O_i$ objects for our type theory example. We begin with
\[
@ -216,13 +266,7 @@
Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
\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 adjunct} of another functor we call $L_{i-1}^i$.
As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will create the two following syntactic sugars for the composed adjunctions.
\[\begin{array}{c}
R_i^j = R_{i}^{i+1} \circ R_{i+1}^j = R_{i}^{j-1} \circ R_{j-1}^{j} = R_{i}^{i+1} \circ ... \circ R_{j-1}^{j}\\
L_i^j = L_{j-1}^{j} \circ L_{i}^{j-1} = L_{i+1}^{j} \circ L_{i}^{i+1} = L_{j-1}^{j} \circ ... \circ L_{i}^{i+1}
\end{array}\]
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 $R_i^j \vdash L_i^j$.
@ -234,17 +278,19 @@
\end{remark}
\subsection{Some Hypotheses}
\subsection{Induction Hypotheses}
In order to build and prove the adjunction, we will state some recurrence invariants that we will prove after having built objects.
\begin{property}[H1]
The canonical morphism
\[
\simpleArrow{R_{i-1}^i X \oplus L_0^{i-1} Y}{\left\{R_{i-1}^i \inj_1^i ; R_{i-1}^i \inj_2^i \circ \eta_{i-1}^i\right\}}{R_{i-1}^i(X \oplus_i L_0^i Y)}
\]
is an isomorphism.
that we will denote as $\en_{i-1}^i$ is an isomorphism.
Its recursive version is the following isomorphism
Its recursive version is the following isomorphism, denoted as $\en_0^i$
\[
\simpleArrow{ R_{0}^i X \oplus_0 Y}{\left\{R_0^i \inj_1^i ; R_0^i \inj_2^i \circ \eta_0^i\right\}}{R_0^i(X \oplus_i L_0^i Y)}
\]
@ -291,16 +337,6 @@
W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}O_i,\dash)}(X,Y), \widetilde{\inj_2} \right)
\]
Where $\Hbar_A$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ defined as
\[
\Hbar_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)}
\]
The morphisms are translated as-is.
\begin{remark}
This functor can be constructed as a lax colimit seeing elements of $A(X)/\Set$ as lax cocones over the diagram $\left[1 \xrightarrow{A(X)} \Set\right]$ in $\Cat$, and seeing elements of $\TSet$ as lax cocones over the diagram with no arrow $\left[\Set \quad \Set\right]$. \inlinetodo{Vérifier ça}
\end{remark}
With $\widetilde{\inj_2}$ being defined by \inlinetodo{Changer les noms des hypothèses H3' et H1r}
\[
\begin{array}{lcl}
@ -351,7 +387,6 @@
% END OF GENERATED LATEX
\end{center}
\subsection{Proof of the adjunction}
We prove that $(W,E)$ make an adjunction showing that there is a natural isomorphism between $\Hom$ sets in both categories.
@ -456,7 +491,7 @@
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-adjunct functor and therefore it preserves colimits; then goes to $X \oplus_i L_0^i Y$.
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 $\varphi_2 : L_0^i Y \to Z$.
@ -754,6 +789,15 @@

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