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\section{Content}
Plan
\subsection{Infinite construction of $\BB_i$}
\[
\BB_i := \left(X : \TSet, \Cstr : (a : S_{i-1}) \to \Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,R_{a-1}^i(\this)) \to X(\UU)\right)
\]
A morphism from $(X,\Cstr)$ to $(X', \Cstr')$ is a morphism from $X$ to $X'$ in $\TSet$ (i.e. a natural transformation $X \implies X'$) which makes the following diagram commute, for all $a$ in $S_{i-1}$.
\diagram{D1}
Identities and compositions are that of the base category $\TSet$, and categorical equalities are trivially derived from the diagram above.
The diagram expresses as
\[
\forall \gamma \in \Hom_{G_{a-1}\Gamma_a,X}, \quad \Cstr'_a(f \circ \gamma) = f_\UU(\Cstr_a \gamma)
\]
\todo{Define properly the use of \this}
\subsection{$H$ functors}
\begin{enumerate}
\item Présentation de ce qu'est un GAT, simple exemple
\item Présentation de la 2-sortification d'un GAT
\item Présentation de la catégorification de Fiore
\item Présentation de la catégorification de Altenkirsch -> Construction des $\BB_i$ récursive
\item Schéma de construction récursif de l'adjonction
\item Formulation de H1 et H3
\item (ANNEXE?) Def de W et de E
\item (ANNEXE?) Preuve de l'adjonction / de la reflexion
\item (ANNEXE?) Preuve des propriétés
\item Définition infinie de $\BB_i$ -> Foncteur $R_0^iG_i$ -> Reflexions sur les catégories pas directes
\end{enumerate}
For every object $X$ of a locally small category $C$, 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{$G$ and $F$ infinite constructions}
\[
G_i(Y) = \left(\sum_{a\in S_i}H_{Y(a)}\left(\lim_{(a/S_i)*}\overline{Y_a}\right), λa.λ\eta.(a,u (\inj_2 \star))\right)
\]
\section{Sort specification}
A Generalized Algebraic Theory (or GAT) and inductive-inductive types are syntaxes describing models.
Both are composed of a sort specification, and eventually a list of constructors.
In this document, we will not ask ourselves about the specificities of both syntaxes. A \enquote{vague} interpretation of them is enough to understand the constructions that follows.
A sort specification is a list of \emph{sort constructors} that are defined with \emph{parameters} with $\Set$ as its codomain.
Here is an example of a classical sort specification for Type Theory.
\[ \Con : \Set \\
\]\[ \Ty : (\Gamma : \Con) \to \Set \\
\]\[ \Tm : (\Gamma : \Delta) \to (A : \Ty \Delta) \to \Set
\]
This specification is to be read as follows:
\begin{center}
There is one sort that is $\Con$
where $u : \left(Y(a) \oplus 1, \inj_1\right) \to (\lim_{(a/S_i)*} \overline{Y_a})$
For every element $\Gamma$ of the sort $\Con$, there is a sort $\Ty \Gamma$
so that $\varphi_{b,f} u (\star) = \eta_\El^b(f)$
\[
F_i(X,\Cstr)(a) = X(p)^{-1}\left(\Cstr_a\left(\Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,X)\right)\right)
\]
\[
F_i(X,\Cstr)(f : a \to b)(X(p)(x); x \in \Cstr_a(\eta)) = \eta_\El^b(f)
\]
\subsection{Hypohesis}
\begin{property}[H3]
\[
F_i(X \oplus L_0^i Y) \cong F_i X
\]
\end{property}
\todo{Do we need to specify that the adjunction is (rtl) $F(inj_1^i)$ ? And prove it ?}
\begin{property}[H3']
\[
\Hom(G_i\Gamma, X) \cong \Hom(G_i\Gamma,X \oplus L_0^i Y)
\]
With left to right isomorphism being $(inj_1^i \circ \dash)$
\end{property}
For every element $\Gamma$ of the sort $\Con$, and every element $A$ of the sort $\Ty \Gamma$, there is a sort $\Tm \Gamma\;F$
\end{center}
We can also add constructors to a sort specification.
For example, for the previous sort specification, one can add the following constructors:
\[
\operatorname{unit} : (\Gamma : \Con) \to \Ty \Gamma
\]
\[
\verb*|_| = \verb*|_|: (\Gamma : \Con) \to (A : \Ty \Gamma) \to \Tm \Gamma A \to \Tm \Gamma A \to \Ty \Gamma
\]
Constructors construct terms and not sorts. A sort specification with or without constructor describes a class of models, which are in the intuitive sense of a model (a model implements \enquote{sort constructors}, regular constructors).
A known process is that one can transform a specification of sorts, into a specification of sorts with two sorts and constructors.
The two sorts are always the same:
\[
\mathcal{O} : \square
\]
\[
\underline{\;\bullet\;} : \mathcal{O} \to \square
\]
Where $\mathcal{O}$ describes the \enquote{sort of sorts}, and $\underline{\;\bullet\;}$ give for every \enquote{sort object} the sort of the \enquote{objects of that sort}.
Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply underline to every sort constructor parameter. For example, the first Type Theory of sort specification becomes that which follows:
\[ \Con : \mathcal{O} \\
\]\[ \Ty : (\Gamma : \underline{\Con}) \to \mathcal{O} \\
\]\[ \Tm : (\Gamma : \underline{\Con}) \to (A : \underline{\Ty \Delta}) \to \mathcal{O}
\]
It is known that this new sorts and constructors specification is equivalent to the former one.
Fiore \cite{Fiore2008} describes \emph{sort specifications} as countable simple direct categories. To every object, we associate a sort, which is built using parameters pointing out of them. We often write this category $S$, and for a sort $a$ in it, the parameters needed to construct the sort are indexed by $(a/S)*$.
As an example, here is the category version of the above specification of sorts. We also add the equality $\Gamma \circ A = \Delta$. This equality describes that the $\Gamma$-parameter i.e. the first parameter of the $\Ty$ sort constructor is the variable $\Delta$. Without this equality, $\Gamma \circ A$ would be another arrow going from $\Tm$, and therefore another parameter on type $\Con$.
\begin{center}
% YADE DIAGRAM B1.json
% GENERATED LATEX
\input{graphs/B1.tex}
% END OF GENERATED LATEX
\end{center}
The category of models over a specification of sorts is then the category of presheaves over the category of Fiore $\left[S,\Set\right]$.
Altenkirsch \cite{Altenkirch2018} has another method to directly construct the category of models of a specification of sorts. His method is more general, but it does not give a \enquote{Tiny and simple} category as Fiore does.
The other advantage of Altenkirsch's method is that they give a way of also describing the models of a constructor specification.
What we will do is to try to make an adjunction between Fiore's category of models $\left[S,\Set\right]$, and Altenkirsch's category of models of the two-sorted GAT.
We will describe how the two categories are constructed in detail.
\subsection{Constructing the categories}
We will construct both categories $S$ and $\BB$ recursively, addings new sorts one by one.
\subsubsection{Fiore's category}
At the same time, we construct the simple category recursively $\emptyset = S_0,S_1,S_2,...$.
In order to construct the $i$-th sort, we use a finite functor $\Gamma_i : \left[S_{i-1} \to \Set\right]$ describing entirely the sort constructor.
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 $(a/S_i)* \equiv \Gamma_a$.TODO C'est sûr la deuxième partie ?
\end{remark}
\subsubsection{Altenkirsch's category}
To start our series of category, we need the category of models of the two-sort specification of sorts ($\mathcal{O}$ and $\underline{\;\bullet\;}$). We do it Fiore's way and we get a simple category that we name $\TT$ with two elements and only one non-trivial arrow between them.
\begin{center}
% YADE DIAGRAM G0.json
% GENERATED LATEX
\input{graphs/G0.tex}
% END OF GENERATED LATEX
\end{center}
The category of models $\TSet$ is also written $\BB_0$ as it is the first step of Altenkirsch's method of adding constructors.
\begin{remark}
Following Altenkirsch's construction of category of models for a sort specification, we end up on the category of families of sets $(X : \Set, Y : X \to \Set)$.
This category is equivalent to $\TSet$ which allows us to give a clearer definition of functors.
\end{remark}
Then, we recursively add constructors, constructing categories $\BB_1$,$\BB_2$, etc.
For the $i$-th constructor, we define the category $\BB_i$ as :
\[
\BB_i := \left(X : \BB_{i-1}, \Cstr_i : \Hom_{\BB_{i-1}} (G_{i-1}\Gamma_i,X) \to (R_0^{i-1}X)_\UU \right)
\]
where $\Gamma_i$ is the functor $S_{i-1} \to \Set$ that describe the sort constructor being processed, and $G_{i-1}$ is the left part of the adjunction $\left[S_{i-1}, \Set\right] \to \BB_{i-1}$ that we are defining recursively at the same time.
A morphism $(X,\Cstr_i) \to (X',\Cstr'_i)$ in $\BB_i$ is a morphism $f : X \to X'$ in $\BB_{i-1}$ such that the following diagram commutes.
\begin{center}
% YADE DIAGRAM D1.json
% GENERATED LATEX
\input{graphs/D1.tex}
% END OF GENERATED LATEX
\end{center}
Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
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.
\[
R_i^j = R_{i}^{i+1} \circ R_{i+1}^j = R_{i}^{j-1} \circ R_{j-1}^{j} = R_{i}^{i+1} \circ ... \circ R_{j-1}^{j}
\]
\[
L_i^j = L_{j-1}^{j} \circ L_{i}^{j-1} = L_{i+1}^{j} \circ L_{i}^{i+1} = L_{j-1}^{j} \circ ... \circ L_{i}^{i+1}
\]
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 know that this category has a coproduct, that we will denote $\oplus_i$ or just $\oplus$ when there is no ambiguity. We will also denote as $\inj_1^i : X \to X \oplus Y$ (resp. $\inj_2^i : Y \to X \oplus Y$) the first (resp. second) injector of the coproduct of $\BB_i$. For every morphisms $f : X \to Z$ and $g : Y \to Z$, we will denote with $\{f;g\}$ the unique morphism from $X \oplus Y$ to $Z$ such that $\{f;g\} \circ \inj^i_1 = f$ and $\{f;g\} \circ \inj^i_2 = g$.
\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 arrow $inj_1 : G_{i-1}\Gamma_i \to G_{i-1}\Gamma_i \oplus L_0^{i-1} y\UU$.
\end{remark}
\subsubsection{Summary}
Here is a graph summarizing the categories and functors.
We have constructed two chains of categories $\BB_0$,$\BB_1$,... and $S_0$,$S_1$,...
The categories $\BB_{i-1}$ and $\BB_{i}$ are in an adjunction written $R_{i-1}^i \vdash L_{i-1}^i$.
We will give in the next part the construction of the adjunction $F_i \vdash G_i$ at the step $i$. The functor $G_{i-1}$ is used in the definition of $\BB_i$, so the two recurrences have to be done at the same time.
\begin{center}
% YADE DIAGRAM G1.json
% GENERATED LATEX
\input{graphs/G1.tex}
% END OF GENERATED LATEX
\end{center}
\subsection{Constructing the adjunction}
\subsubsection{Hypotheses}
In order to build and prove the adjunction, we will state hypotheses that we will progressively prove during our building of $\BB_i$, $F_i$, and $G_i$.
\begin{property}[H1]
\[
R_{i-1}^i(X \oplus_i L_0^i Y) \cong R_{i-1}^i X \oplus L_0^{i-1} Y
\simpleArrow{R_{i-1}^i X \oplus L_0^{i-1} Y}{\left\{R_{i-1}^i \inj_1^i ; R_{i-1}^i \inj_2^i \circ \eta_{i-1}^i\right\}}{R_{i-1}^i(X \oplus_i L_0^i Y)}
\]
is an equivalence.
\end{property}
\begin{property}[H1r]
\[
R_{0}^i(X \oplus_i L_0^i Y) \cong R_{0}^i X \oplus_{i-1} L_0^{i-1} Y
\simpleArrow{ R_{0}^i X \oplus_{i-1} Y}{\left\{R_{i-1}^i \inj_1^i ; \eta_{i-1}^i\right\}}{R_{i-1}^i(X \oplus_i L_0^i Y)}
\]
We use the sum that makes this equivalence to be an equality.
is an equivalence.
This property is proven easily by recursion over previous property H1.
\end{property}
\subsection{Notations}
\[
R_i^j = R_{i}^{i+1} \circ R_{i+1}^j = R_{i}^{j-1} \circ R_{j-1}^{j} = R_{i}^{i+1} \circ ... \circ R_{j-1}^{j}
\]
\[
L_i^j = L_{j-1}^{j} \circ L_{i}^{j-1} = L_{i+1}^{j} \circ L_{i}^{i+1} = L_{j-1}^{j} \circ ... \circ L_{i}^{i+1}
\]
\begin{property}[H3]
\[
\simpleArrow{F_i X}{F(\inj_1^i)}{F_i(X \oplus L_0^i Y)}
\]
is an equivalence.
\end{property}
In any category with a coproduct, for every morphisms $f : X \to Z$ and $g : Y \to Z$, we denote with $\{f;g\}$ the unique morphism from $X \oplus Y$ to $Z$ such that $\{f;g\} \circ \inj_1 = f$ and $\{f;g\} \circ \inj_2 = g$.
\subsection{Recursive definition of $\BB_i$}
\begin{property}[H3']
\[
\simpleArrow{\Hom(G_i\Gamma, X)}{(inj_1^i \circ \dash)}{\Hom(G_i\Gamma,X \oplus L_0^i Y)}
\]
is an equivalence.
This proven with the adjunction property of $F_i \vdash G_i$ and H3, as w have that
\[
\Hom(G_i \Gamma, X) \cong \Hom(\Gamma, F_i X) \cong \Hom(\Gamma, F_i(X \oplus L_0^i Y)) \cong \Hom(G_i \Gamma, X \oplus L_0^i Y)
\]
\end{property}
\[
\BB_i := \left(X : \BB_{i-1}, \Cstr_i : \Hom_{\BB_{i-1}} (G_{i-1}\Gamma_i,X) \to (R_0^{i-1}X)_\UU \right)
\]
A morphism $(X,\Cstr_i) \to (X',\Cstr'_i)$ is a morphism $f : X \to X'$ in $\BB_{i-1}$ such that the following diagram commutes.
\diagram{D1 $[a \mapsfrom i]$}
Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above.
\subsection{W definition}
\subsubsection{Decomposing the proof}
We define a functor $W : \displaystyle\prod_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,X)) \to \BB_{i}$
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.
\begin{center}
% YADE DIAGRAM G2.json
% GENERATED LATEX
\input{graphs/G2.tex}
% END OF GENERATED LATEX
\end{center}
The category $\left[S_i, \Set\right]$ is seen as the category $\left[S_{i-1},\Set\right]$ to which we have added an object along with morphisms described by $\Gamma_i$. The morphisms we added to the object $X$ have the shape of the slice category of the set $\Hom(\Gamma_i,X)$.
The first part $G_{i-1} \times \id \dashv F_{i-1} \times \id$ is proven and defined as an adjunction from the last step of the recurrence.
We will now define the two functors.
\subsubsection{W definition}
We define a functor $W : \displaystyle\sum_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,X)) \to \BB_{i}$
\[
W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}\Gamma_i,\dash)}(X,Y), \widetilde{\inj_2} \right)
W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}\Gamma_i,\dash)}(X,Y), \widetilde{\inj_2} \right)
\]
With $\widetilde{\inj_2}$ being defined by
@ -147,40 +295,58 @@
The action on a morphism $(g,h)$ from $(X,Y)$ to $(X',Y')$ is the following:
\[
W(g,h) := \left(g \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}\Gamma_i,\dash)}(g,h)\right)
W(g,h) := \left(g \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}\Gamma_i,\dash)}(g,h)\right)
\]
It is indeed a morphism from $\BB_{i}$ as it makes the following diagram commute.
\diagram{D2}
\subsection{E definition}
We define $E : \BB_{i} \to \displaystyle\prod_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,X))$
\begin{center}
% YADE DIAGRAM D2.json
% GENERATED LATEX
\input{graphs/D2.tex}
% END OF GENERATED LATEX
\end{center}
\subsubsection{E definition}
We define $E : \BB_{i} \to \displaystyle\sum_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,X))$
\[
E(X) = (R_{i-1}^i X, (A,h))
E(X) = (R_{i-1}^i X, (A,h))
\]
Where $(A,h)$ is defined as the following pullback:
\diagram{D3a}
\begin{center}
% YADE DIAGRAM D3a.json
% GENERATED LATEX
\input{graphs/D3a.tex}
% END OF GENERATED LATEX
\end{center}
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:
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):
\diagram{D3b}
\begin{center}
% YADE DIAGRAM D3b.json
% GENERATED LATEX
\input{graphs/D3b.tex}
% END OF GENERATED LATEX
\end{center}
\subsubsection{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.
We want to construct for each $(X,Y)$ in $\displaystyle\prod_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,X))$ and $Z$ in $\BB_i$, an isomorphism $\phi_{XYZ}$.
\[
\phi_{XYZ} : \Hom(W(X,Y),Z) \to \Hom((X,Y),E(Z))
\]
I will give the construction of the isomorphisms and its inverse, the proofs are given in \autoref{apx:phi-WE-isnat}.
\paragraph{Constructing $\phi_{XYZ}$}
\subsection{W E adjunction}
We have an adjunction $W \dashv E$.
Let $(X,Y)$ be in $\displaystyle\prod_{X : \BB_{i-1}} (\Set/\Hom_{\BB_{i-1}}(G_{i-1}\Gamma_i,X))$ and $Z$ be in $\BB_i$.
We will construct a natural isomorphism.
\[
\phi_{XYZ} : \Hom(W(X,Y),Z) \to \Hom((X,Y),E(Z))
\]
\subsubsection{Constructing $\phi_{XYZ}$}
Let $f$ be in $\Hom(W(X,Y),Z)$.
We want to construct $\phi_{XYZ}(f) : (X,Y) \to E(Z)$.
@ -188,104 +354,45 @@
The second component is defined through the universal property of the pullback defined by $E(Z)$ according to the following diagram:
\diagram{D6}
\begin{center}
% YADE DIAGRAM D6.json
% GENERATED LATEX
\input{graphs/D6.tex}
% END OF GENERATED LATEX
\end{center}
\subsubsection{Constructing $\phi^{-1}_{XYZ}$}
\paragraph{Constructing $\phi^{-1}_{XYZ}$}
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:
\[
\phi^{-1}_{XYZ}(g,h) = \left\{g ; \eta_0^i \circ L_0^{i-1} \square \right\}
\phi^{-1}_{XYZ}(g,h) := \left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\}
\]
Where $\square$ is a morphism $\Hbar (X,Y) \to R_0^i Z$ defined by the following diagram:
\diagram{D7}
\begin{center}
% YADE DIAGRAM D7.json
% GENERATED LATEX
\input{graphs/D7.tex}
% END OF GENERATED LATEX
\end{center}
This is indeed a morphism of $\BB_i$ from $W(X,Y)$ to $Z$ as it makes the following diagram commute
\diagram{D8}
\begin{center}
% YADE DIAGRAM D8.json
% GENERATED LATEX
\input{graphs/D8.tex}
% END OF GENERATED LATEX
\end{center}
We now have to show that $\phi_{XYZ}$ and $\phi_{XYZ}^{-1}$ do make a natural isomorphism.
We show in \autoref{apx:phi-WE-isnat} that $\phi_{XYZ}$ and $\phi_{XYZ}^{-1}$ do indeed make a natural isomorphism.
\subsubsection{Composition $\phi_{XYZ} \circ \phi_{XYZ}^{-1}$}
The first component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ is
\[
R_{i-1}^i(\left\{g ; \eta_0^i \circ L_0^{i-1} \square \right\}) \circ \inj_1^{i-1} = \left\{g ; \eta_0^i \circ L_0^{i-1} \square \right\} \circ \inj_1^{i-1} = g
\]
The second component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ follows the following diagram
\diagram{D9}
The diagram commutes as the following equality holds:
\[
\left(\left\{g ; \eta_0^i \circ L_0^{i-1} \square \right\} \circ \inj^{i-1}_2\right)_\El = \left(\eta_0^i \circ L_0^{i-1} \square\right)_\El = (\eta_0^i)_\El \circ (L_0^{i-1} \square)_\El = \square_El = \pi_1 \circ h
\]
\todo{Justify $(\eta_0^i)_\El = id_{\BB_i}$ and $(L_0^i(f))_\El = f_\El$}
So, as the square is a pullback, we get the complete equality
\[
\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h)) = (g,h)
\]
\subsubsection{Composition $\phi_{XYZ}^{-1} \circ \phi_{XYZ}$}
Now, the converse composition is
\[
\phi_{XYZ}^{-1} (\phi_{XYZ}(f)) = \left\{R_{i-1}^i f \circ \inj_1^{i-1} ; \eta_0^i \circ L_0^{i-1} \square \right\}
\]
where $\square$ follows the following diagram
\diagram{D10}
We want to show that $\phi_{XYZ}^{-1} (\phi_{XYZ}(f))$. By definition of $\{;\}$ in $\BB_1$, it suffices to show that $\eta_0^i \circ L_0^{i-1} \square = R_{i-1}^i f \circ \inj_1^{i-1}$.
So it suffices to show that
\[
\square = R_0^{i-1}(R_{i-1}^i f \circ \inj_2^{i-1}) \circ \varepsilon_0^{i-1} = R_0^i f \circ \left(R_0^{i-1} \inj_2^{i-1} \circ \varepsilon_0^{i-1}\right) = R_0^i f \circ \inj_2^0
\]
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
\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 $R_0^{i-1} \inj_2^{i-1} \circ \varepsilon_0^{i-1} = \inj_2^0$}
\subsubsection{Naturality}
We want to show that the following diagram commutes, for any objects $X$,$Y$,$Z$,$X'$,$Y'$,$Z'$ and morphisms $f$,$g$,$h$.
\diagram{D10.0}
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.
\[\begin{array}{rcl}
\phi_{XYZ}(f \circ \ii \circ W(g,h))_1
&=& R_{i-1}^i\left(f \circ \ii \circ (g \oplus L_0^{i-1} \Hbar(g,h))^+\right) \circ \inj_1^{i-1} \\
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \left\{\inj_1^{i-1} \circ g; \dots\right\} \circ \inj_1^{i-1} \\
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \inj_1^{i-1} \circ g \\
&=& \left[E(f) \circ \phi_{X'Y'Z'}(\ii) \circ (g,h)\right]_1
\end{array}\]
The second components are defined as described by the following diagram
\diagram{D10}
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.
\subsection{$F \vdash G$ make a reflexion}
\subsection{Proof of H1 - Sum definition}
@ -309,26 +416,114 @@
It is a morphism of $\BB_i$ as it makes the following diagram commute:
\diagram{D4}
\begin{center}
% YADE DIAGRAM D4.json
% GENERATED LATEX
\input{graphs/D4.tex}
% END OF GENERATED LATEX
\end{center}
The second injector is defined as follows:
\[
inj_2^i := (\eta_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
inj_2^i := (\varepsilon_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
\]
Where $\eta_i$ 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 $X$.
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 $X \oplus_i L_0^i Y$.
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 $\{\varphi_1 ; \varphi_2\}_{i } = \{R_{i-1}^i \varphi_1 ; R_{i-1}^i \varphi_2 \circ \varepsilon^i_{L_0^{i-1} Y}\}_{i-1}$ a morphism of $\BB_i$ as it makes the following diagram commute:
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:
\diagram{D5}
\begin{center}
% YADE DIAGRAM D5.json
% GENERATED LATEX
\input{graphs/D5.tex}
% END OF GENERATED LATEX
\end{center}
\todo{Justifier $R_{i-1}^i(\eta_i \oplus_i \id_{L_0^i Y}) = R_{i-1}^i \eta_i \oplus_{i-1} \id_{L_0^{i-1} Y}$ (with H1 ?)}
\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 ?)}
\subsection{Proof of H3}
\subsection{Infinite construction of $\BB_i$}
\[
\BB_i := \left(X : \TSet, \Cstr : (a : S_{i-1}) \to \Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,R_{a-1}^i(\this)) \to X(\UU)\right)
\]
A morphism from $(X,\Cstr)$ to $(X', \Cstr')$ is a morphism from $X$ to $X'$ in $\TSet$ (i.e. a natural transformation $X \implies X'$) which makes the following diagram commute, for all $a$ in $S_{i-1}$.
\begin{center}
% YADE DIAGRAM D1.json
% GENERATED LATEX
\input{graphs/D1.tex}
% END OF GENERATED LATEX
\end{center}
Identities and compositions are that of the base category $\TSet$, and categorical equalities are trivially derived from the diagram above.
The diagram expresses as
\[
\forall \gamma \in \Hom_{G_{a-1}\Gamma_a,X}, \quad \Cstr'_a(f \circ \gamma) = f_\UU(\Cstr_a \gamma)
\]
\todo{Define properly the use of \this}
\subsection{$G$ and $F$ infinite constructions}
\[
G_i(Y) = \left(\sum_{a\in S_i}H_{Y(a)}\left(\lim_{(a/S_i)*}\overline{Y_a}\right), λa.λ\eta.(a,u (\inj_2 \star))\right)
\]
where $u : \left(Y(a) \oplus 1, \inj_1\right) \to (\lim_{(a/S_i)*} \overline{Y_a})$
so that $\varphi_{b,f} u (\star) = \eta_\El^b(f)$
\[
F_i(X,\Cstr)(a) = X(p)^{-1}\left(\Cstr_a\left(\Hom_{\BB_{a-1}}(G_{a-1}\Gamma_a,X)\right)\right)
\]
\[
F_i(X,\Cstr)(f : a \to b)(X(p)(x); x \in \Cstr_a(\eta)) = \eta_\El^b(f)
\]
\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{Overview}
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.
\section{Summary}
@ -348,5 +543,106 @@
\addtocontents{toc}{\protect\setcounter{tocdepth}{-1}}
\appendixpage
\section{$W \dashv E$ adjunction}
\label{apx:phi-WE-isnat}
\subsection{Composition $\phi_{XYZ} \circ \phi_{XYZ}^{-1}$}
The first component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ is
\[
R_{i-1}^i(\left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\}) \circ \inj_1^{i-1} = \left\{g ; \varepsilon_0^i \circ L_0^{i-1} \square \right\} \circ \inj_1^{i-1} = g
\]
The second component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ follows the following diagram
\begin{center}
% YADE DIAGRAM D9.json
% GENERATED LATEX
\input{graphs/D9.tex}
% 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 $(L_0^i(f))_\El = f_\El$}
So, as the square is a pullback, we get the complete equality
\[
\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h)) = (g,h)
\]
\subsection{Composition $\phi_{XYZ}^{-1} \circ \phi_{XYZ}$}
Now, the converse composition is
\[
\phi_{XYZ}^{-1} (\phi_{XYZ}(f)) = \left\{R_{i-1}^i f \circ \inj_1^{i-1} ; \varepsilon_0^i \circ L_0^{i-1} \square \right\}
\]
where $\square$ follows the following diagram
\begin{center}
% YADE DIAGRAM D10.json
% GENERATED LATEX
\input{graphs/D10.tex}
% 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
\[
\square = R_0^{i-1}(R_{i-1}^i f \circ \inj_2^{i-1}) \circ \eta_0^{i-1} = R_0^i f \circ \left(R_0^{i-1} \inj_2^{i-1} \circ \eta_0^{i-1}\right) = R_0^i f \circ \inj_2^0
\]
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
\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 $f$,$g$,$h$.
\begin{center}
% YADE DIAGRAM D10.0.json
% GENERATED LATEX
\input{graphs/D10.0.tex}
% 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 first look at the first component of the result morphism.
\[\begin{array}{rcl}
\phi_{XYZ}(f \circ \ii \circ W(g,h))_1
&=& R_{i-1}^i\left(f \circ \ii \circ (g \oplus L_0^{i-1} \Hbar(g,h))^+\right) \circ \inj_1^{i-1} \\
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \left\{\inj_1^{i-1} \circ g; \dots\right\} \circ \inj_1^{i-1} \\
&=& R_{i-1}^i f \circ R_{i-1}^i \ii \circ \inj_1^{i-1} \circ g \\
&=& \left[E(f) \circ \phi_{X'Y'Z'}(\ii) \circ (g,h)\right]_1
\end{array}\]
The second components are defined as described by the following diagram
\begin{center}
% YADE DIAGRAM D11.json
% GENERATED LATEX
\input{graphs/D11.tex}
% 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.
\end{document}

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13
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@ -39,6 +39,7 @@
\usepackage{pdfpages}
\usepackage{lipsum}
\usepackage{newunicodechar}
\usepackage{yade}
\usepackage[textheight=0.75\paperheight]{geometry}
@ -69,6 +70,7 @@
\newtheorem{theorem}{Théorème}
\newtheorem{definition}{Definition}
\newtheorem{property}{Propriété}
\newtheorem{remark}{Remark}
\newcounter{rule}
\addto\extrasfrench{%
@ -103,6 +105,9 @@
\newcommand\UU{{\ensuremath{\mathcal{U}}}}
\newcommand\El{{\ensuremath{\operatorname{\mathcal{E}l}}}}
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\right]
}}}
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15
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@ -0,0 +1,15 @@
\newcommand\ensuremath[1]{#1}
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960
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@ -0,0 +1,960 @@
% Copyright 2022 by Tom Hirschowitz
%
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\@ifnextchar[{\twocell@ii[#1]}{\twocell@ii[#1][#1]}%
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(a) ++ (\p5) node[coordinate] (x) {} %%
(c) ++ (\p6) node[coordinate] (y) {} %%
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\deuxcellule{#1}{\IfNoValueTF{#2}{#1}{#2}}{#5}{#6}{\IfNoValueTF{#7}{#5}{#7}}{#8}{}{bend left={#3},celllr=#4,label={[above right]{$\scriptstyle #9$}}}}
\DeclareDocumentCommand{\twocellbl}{O{.4} o D(){30} > { \SplitArgument { 1 } { , } } D<>{0.5,0.5} m m o m m}{%
\deuxcellule{#1}{\IfNoValueTF{#2}{#1}{#2}}{#5}{#6}{\IfNoValueTF{#7}{#5}{#7}}{#8}{}{bend right={#3},celllr=#4,label={[below left]{$\scriptstyle #9$}}}}
% Two cell bent right
\def\twocellrb{%
\@ifnextchar[{\twocellrb@i}{\twocellrb@i[.4]}%
}
\def\twocellrb@i[#1]{%
\@ifnextchar[{\twocellrb@ii[#1]}{\twocellrb@ii[#1][#1]}%
}
\def\twocellrb@ii[#1][#2]{%
\deuxcellulerb{#1}{#2}%
}
\newcommand{\deuxcellulerb}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labeld={#6}}}
\def\twocellra{%
\@ifnextchar[{\twocellra@i}{\twocellra@i[.4]}%
}
\def\twocellra@i[#1]{%
\@ifnextchar[{\twocellra@ii[#1]}{\twocellra@ii[#1][#1]}%
}
\def\twocellra@ii[#1][#2]{%
\deuxcellulera{#1}{#2}%
}
\newcommand{\deuxcellulera}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelu={#6}}}
\def\twocellral{%
\@ifnextchar[{\twocellral@i}{\twocellral@i[.4]}%
}
\def\twocellral@i[#1]{%
\@ifnextchar[{\twocellral@ii[#1]}{\twocellral@ii[#1][#1]}%
}
\def\twocellral@ii[#1][#2]{%
\deuxcelluleral{#1}{#2}%
}
\newcommand{\deuxcelluleral}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelal={#6}}}
\def\twocellro{%
\@ifnextchar[{\twocellro@i}{\twocellro@i[.4]}%
}
\def\twocellro@i[#1]{%
\@ifnextchar[{\twocellro@ii[#1]}{\twocellro@ii[#1][#1]}%
}
\def\twocellro@ii[#1][#2]{%
\deuxcellulero{#1}{#2}%
}
\newcommand{\deuxcellulero}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelo={#6}}}
\def\twocellrl{%
\@ifnextchar[{\twocellrl@i}{\twocellrl@i[.4]}%
}
\def\twocellrl@i[#1]{%
\@ifnextchar[{\twocellrl@ii[#1]}{\twocellrl@ii[#1][#1]}%
}
\def\twocellrl@ii[#1][#2]{%
\deuxcellulerl{#1}{#2}%
}
\newcommand{\deuxcellulerl}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labell={#6}}}
\def\twocellrr{%
\@ifnextchar[{\twocellrr@i}{\twocellrr@i[.4]}%
}
\def\twocellrr@i[#1]{%
\@ifnextchar[{\twocellrr@ii[#1]}{\twocellrr@ii[#1][#1]}%
}
\def\twocellrr@ii[#1][#2]{%
\deuxcellulerr{#1}{#2}%
}
\newcommand{\deuxcellulerr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelr={#6}}}
\def\twocellrbr{%
\@ifnextchar[{\twocellrbr@i}{\twocellrbr@i[.4]}%
}
\def\twocellrbr@i[#1]{%
\@ifnextchar[{\twocellrbr@ii[#1]}{\twocellrbr@ii[#1][#1]}%
}
\def\twocellrbr@ii[#1][#2]{%
\deuxcellulerbr{#1}{#2}%
}
\newcommand{\deuxcellulerbr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelbr={#6}}}
% Two cell bent left
\def\twocelllb{%
\@ifnextchar[{\twocelllb@i}{\twocelllb@i[.4]}%
}
\def\twocelllb@i[#1]{%
\@ifnextchar[{\twocelllb@ii[#1]}{\twocelllb@ii[#1][#1]}%
}
\def\twocelllb@ii[#1][#2]{%
\deuxcellulelb{#1}{#2}%
}
\newcommand{\deuxcellulelb}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labeld={#6}}}
\def\twocellla{%
\@ifnextchar[{\twocellla@i}{\twocellla@i[.4]}%
}
\def\twocellla@i[#1]{%
\@ifnextchar[{\twocellla@ii[#1]}{\twocellla@ii[#1][#1]}%
}
\def\twocellla@ii[#1][#2]{%
\deuxcellulela{#1}{#2}%
}
\newcommand{\deuxcellulela}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelu={#6}}}
\def\twocelllal{%
\@ifnextchar[{\twocelllal@i}{\twocelllal@i[.4]}%
}
\def\twocelllal@i[#1]{%
\@ifnextchar[{\twocelllal@ii[#1]}{\twocelllal@ii[#1][#1]}%
}
\def\twocelllal@ii[#1][#2]{%
\deuxcellulelal{#1}{#2}%
}
\newcommand{\deuxcellulelal}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelal={#6}}}
\def\twocelllo{%
\@ifnextchar[{\twocelllo@i}{\twocelllo@i[.4]}%
}
\def\twocelllo@i[#1]{%
\@ifnextchar[{\twocelllo@ii[#1]}{\twocelllo@ii[#1][#1]}%
}
\def\twocelllo@ii[#1][#2]{%
\deuxcellulelo{#1}{#2}%
}
\newcommand{\deuxcellulelo}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelo={#6}}}
\def\twocellll{%
\@ifnextchar[{\twocellll@i}{\twocellll@i[.4]}%
}
\def\twocellll@i[#1]{%
\@ifnextchar[{\twocellll@ii[#1]}{\twocellll@ii[#1][#1]}%
}
\def\twocellll@ii[#1][#2]{%
\deuxcellulell{#1}{#2}%
}
\newcommand{\deuxcellulell}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labell={#6}}}
\def\twocelllr{%
\@ifnextchar[{\twocelllr@i}{\twocelllr@i[.4]}%
}
\def\twocelllr@i[#1]{%
\@ifnextchar[{\twocelllr@ii[#1]}{\twocelllr@ii[#1][#1]}%
}
\def\twocelllr@ii[#1][#2]{%
\deuxcellulelr{#1}{#2}%
}
\newcommand{\deuxcellulelr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelr={#6}}}
\def\twocelllbr{%
\@ifnextchar[{\twocelllbr@i}{\twocelllbr@i[.4]}%
}
\def\twocelllbr@i[#1]{%
\@ifnextchar[{\twocelllbr@ii[#1]}{\twocelllbr@ii[#1][#1]}%
}
\def\twocelllbr@ii[#1][#2]{%
\deuxcellulelbr{#1}{#2}%
}
\newcommand{\deuxcellulelbr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelbr={#6}}}
\newbox\xrat@below
\newbox\xrat@above
\newcommand{\Xarrow}[3][]{%
\setbox\xrat@below=\hbox{\ensuremath{\scriptstyle #2}}%
\setbox\xrat@above=\hbox{\ensuremath{\scriptstyle #3}}%
\pgfmathsetlengthmacro{\xrat@len}{max(\wd\xrat@below,\wd\xrat@above)+.6em}%
\mathrel{\tikz [#1,baseline=-.75ex]
\draw (0,0) -- node[below=-2pt] {\box\xrat@below}
node[above=-2pt] {\box\xrat@above}
(\xrat@len,0) ;}}
\newcommand{\xarrow}[2][]{%
\setbox\xrat@above=\hbox{\ensuremath{\scriptstyle #2\ }}%
\pgfmathsetlengthmacro{\xrat@len}{\wd\xrat@above+.8em}%
\mathrel{\tikz [baseline=-.75ex]
\draw (0,0) edge[#1] node[above=-2pt] {\box\xrat@above}
(\xrat@len,0) ;}}
\makeatother
\newenvironment{net}{\begin{tikzpicture}[baseline=(current bounding box.center),text depth=.2em,text height=.8em,inner sep=1pt]}{\end{tikzpicture}}
\newcommand{\ssf}[2]{%
\inetatom(#1){#2}%
}
\newcommand{\gimpll}{\ssf{Rien}{\impll}}
\newcommand{\gtens}{\ssf{Rien}{\tens}}
\newcommand{\gimpllof}[1]{\ssf{#1}{\impll}}
\newcommand{\gtensof}[1]{\ssf{#1}{\tens}}
\newcommand{\gun}[1]{\ssf{#1}{\un}}
\newcommand{\rpare}[1]{\node[text depth=.2em,text height=.8em,inner sep=0pt] (#1) {\ensuremath{)}};}
\newcommand{\lpare}[1]{\node[text depth=.2em,text height=.8em,inner sep=0pt] (#1) {\ensuremath{(}};}
\newcommand{\point}[1]{\node[right=0cm of #1] {.} ; }
\newcommand{\virgule}[1]{\node[right=0cm of #1] {,} ; }
\newcommand{\poing}{\ssf{}{.}}
\newcommand{\rieng}{\ssf{}{\,}}
\pgfdeclaredecoration{single line}{initial}{
\state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}}
\state{final}{\pgfpathlineto{\pgfpointorigin}}
}
\pgfdeclaredecoration{single line backwards}{initial}{
\state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}}
\state{final}{\pgfpathlineto{\pgfpointorigin}}
}
\tikzset{
raise line/.style={
decoration={single line, raise=#1}, decorate
}
}
\tikzset{
mod/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {
\path[draw,-] (0,-3pt) -- (0,3pt);
}}}}}
\tikzset{
label/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\node[inner sep=2pt,outer sep=0] #1 ;}
}}}}
\tikzset{
labelo/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\node[circle,inner sep=0pt,fill=white] {$\scriptstyle #1$} ;}
}}}}
\tikzset{
labelon/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\node[inner sep=1pt,fill=white] {$\scriptstyle #1$} ;}
}}}}
\tikzset{
labelonb/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\node[inner sep=0pt,fill={bg}] {$\scriptstyle #1$} ;}
}}}}
\tikzset{
labelat/.style 2 args={postaction={
decorate,
decoration={
markings,
mark=at position #2 with {\node[inner sep=2pt] #1 ;}
}}}}
\tikzset{
labeloat/.style 2 args={postaction={
decorate,
decoration={
markings,
mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;}
}}}}
\tikzset{
labelonat/.style 2 args={postaction={
decorate,
decoration={
markings,
mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;}
}}}}
\tikzset{diagnode/.style={anchor=base,inner sep=5pt,outer sep=0pt}}
\tikzset{diag/.style 2 args=%
{
matrix of math nodes,ampersand replacement=\&, %
text height=1.2ex, text depth=0.25ex, %
row sep={#1 cm,between borders}, %
column sep={#2 cm,between borders}}
}%
\tikzset{diagorigins/.style 2 args=%
{matrix of math nodes,ampersand replacement=\&, %
row sep={#1 cm,between origins}, %
column sep={#2 cm,between origins}}
}%
\tikzset{stringdiag/.style 2 args=%
{nodes={inner sep=1pt,outer sep=0pt},%
ampersand replacement=\&,%
row sep={#1 cm,between origins}, %
column sep={#2 cm,between origins}%
}}%
\newcommand{\diaggrandhauteur}{1}
\newcommand{\diaggrandlargeur}{2}
\newcommand{\diagpetithauteur}{.5}
\newcommand{\diagpetitlargeur}{1.5}
\tikzset{organigram/.style 2 args={matrix of nodes,ampersand replacement=\&, %
text height=1.7ex, text depth=0.25ex, %
row sep={#1 cm,between origins}, %
column sep={#2 cm,between origins} }}
\tikzset{graphe/.style 2 args={matrix of math nodes,ampersand replacement=\&, %
row sep={#1 cm,between origins}, %
column sep={#2 cm,between origins}, %
inner sep=-.1ex}} %
\tikzset{
two/.style 2 args={postaction={
decorate,
decoration={
markings,
mark=at position .5 with \node (#1) [#2] {} ;
}}}}
\tikzset{
twocenter/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with \node (#1) {} ;
}}}}
\tikzset{
twoon/.style 2 args={twocenter={#1},label={{$\scriptstyle #2$}}}
}
\tikzset{
twoo/.style={two={on},label={{$\scriptstyle #1$}}}
}
\tikzset{
twol/.style={two={l}{right},label={[left]{$\scriptstyle #1$}}}
}
\tikzset{
twoleft/.style 2 args={two={#1}{right},label={[left]{$\scriptstyle #2$}}}
}
\tikzset{
twor/.style={two={r}{left},label={[right]{$\scriptstyle #1$}}}
}
\tikzset{
tworight/.style 2 args={two={#1}{left},label={[right]{$\scriptstyle #2$}}}
}
\tikzset{
twou/.style={two={u}{below},label={[above]{$\scriptstyle #1$}}}
}
\tikzset{
twoa/.style={two={a}{below},label={[above]{$\scriptstyle #1$}}}
}
\tikzset{
twoup/.style 2 args={two={#1}{below},label={[above]{$\scriptstyle #2$}}}
}
\tikzset{
twoabove/.style 2 args={two={#1}{below},label={[above]{$\scriptstyle #2$}}}
}
\tikzset{
twod/.style={two={d}{above},label={[below]{$\scriptstyle #1$}}}
}
\tikzset{
twob/.style={two={b}{above},label={[below]{$\scriptstyle #1$}}}
}
\tikzset{
twodown/.style 2 args={two={#1}{above},label={[below]{$\scriptstyle #2$}}}
}
\tikzset{
twobelow/.style 2 args={two={#1}{above},label={[below]{$\scriptstyle #2$}}}
}
\tikzset{
twoal/.style={two={al}{right},label={[above left]{$\scriptstyle #1$}}}
}
\tikzset{
twoaboveleft/.style 2 args={two={#1}{above},label={[above left]{$\scriptstyle #2$}}}
}
\tikzset{
twoar/.style={two={ar}{left},label={[above right]{$\scriptstyle #1$}}}
}
\tikzset{
twoaboveright/.style 2 args={two={#1}{above},label={[above right]{$\scriptstyle #2$}}}
}
\tikzset{
twobr/.style={two={br}{left},label={[below right]{$\scriptstyle #1$}}}
}
\tikzset{
twobelowright/.style 2 args={two={#1}{left},label={[below right]{$\scriptstyle #2$}}}
}
\tikzset{
twobl/.style={two={bl}{right},label={[below left]{$\scriptstyle #1$}}}
}
\tikzset{
twobelowleft/.style 2 args={two={#1}{right},label={[below left]{$\scriptstyle #2$}}}
}
\tikzset{
twol/.style={two={l}{right},label={[left]{$\scriptstyle #1$}}}
}
\tikzset{
labell/.style={label={[left]{$\scriptstyle #1$}}}
}
\tikzset{
labellat/.style 2 args={labelat={[left]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labelr/.style={label={[right]{$\scriptstyle #1$}}}
}
\tikzset{
labelrat/.style 2 args={labelat={[right]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labelar/.style={label={[above right]{$\scriptstyle #1$}}}
}
\tikzset{
labelarat/.style 2 args={labelat={[above right]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labelbr/.style={label={[below right]{$\scriptstyle #1$}}}
}
\tikzset{
labelbrat/.style 2 args={labelat={[below right]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labelu/.style={label={[above]{$\scriptstyle #1$}}}
}
\tikzset{
labeluat/.style 2 args={labelat={[above]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labela/.style={label={[above]{$\scriptstyle #1$}}}
}
\tikzset{
labelaat/.style 2 args={labelat={[above]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
loina/.style={label={[above=.5em]{$\scriptstyle #1$}}}
}
\tikzset{
labeld/.style={label={[below]{$\scriptstyle #1$}}}
}
\tikzset{
labelb/.style={label={[below]{$\scriptstyle #1$}}}
}
\tikzset{
labelbat/.style 2 args={labelat={[below]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
loinb/.style={label={[below=.5em]{$\scriptstyle #1$}}}
}
\tikzset{
labelal/.style={label={[above left]{$\scriptstyle #1$}}}
}
\tikzset{
labelalat/.style 2 args={labelat={[above left]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labelbl/.style={label={[below left]{$\scriptstyle #1$}}}
}
\tikzset{
labelblat/.style 2 args={labelat={[below left]{$\scriptstyle #1$}}{#2}}
}
\tikzset{
labellat/.style 2 args={labelat={[left]{$\scriptstyle #1$}}{#2}}
}
\newcommand{\cs}[2][draw,->]{ %
\path[#1] (#2-1-1) -- (#2-1-2) ; %
\path[#1] (#2-1-3) -- (#2-1-2) ; %
}
\newcommand{\cospan}{\cs{m}}
\newcommand{\vdoublecs}[2][draw,->]{
\foreach \i in {1,2,3} %
{%
\path[#1] (#2-1-\i) -- (#2-2-\i) ;%
\path[#1] (#2-3-\i) -- (#2-2-\i) ; %
} ; %
}
\newcommand{\hdoublecs}[2][draw,->]{
\foreach \i in {1,2,3} %
{%
\path[#1] (#2-\i-1) -- (#2-\i-2) ;%
\path[#1] (#2-\i-3) -- (#2-\i-2) ; %
} ; %
}
\newcommand{\vdoubleisos}[1]{
\foreach \i in {1,2,3} %
{%
\isopath{#1-1-\i}{#1-2-\i} %
\isopath{#1-2-\i}{#1-3-\i} %
} ; %
}
\newcommand{\hdoubleisos}[1]{
\foreach \i in {1,2,3} %
{%
\isopath{#1-\i-1}{#1-\i-2} %
\isopath{#1-\i-2}{#1-\i-3} %
} ; %
}
\newcommand{\doublecs}[2][draw,->]{%
\vdoublecs[#1]{#2} %
\hdoublecs[#1]{#2} %
}
\newcommand{\doublecospan}{\doublecs{m}}
\newcommand{\sq}[4]{%
(m-1-1) edge[twou={#1}] (m-1-2) %
(m-1-1) edge[twol={#2}] (m-2-1) %
(m-1-2) edge[twor={#3}] (m-2-2) %
(m-2-1) edge[twod={#4}] (m-2-2) %
}
\newcommand{\sqpath}[6]{
\draw[->,#6,rounded corners]
(#1) -- +(#2:#3ex) -- node(#4) {} ($(#5) + (#2:#3ex)$) -- (#5) %
; %
}
\newcommand{\celltoangle}[5]{
\path[draw] ($(#1)+(#2:#3ex)$) edge[celllr={0}{0},#5] ($(#1)+(#2:#4ex)$) ; %
}
\newcommand{\pbkdefault}{1.4em}
\newcommand{\pbkmargin}{1pt}
\DeclareDocumentCommand{\pullbackk}{O{\pbkdefault} O{\pbkdefault} D(){2pt} m m m m}{%
\node[coordinate] (a) at (#4) {} ; %
\node[coordinate] (b) at (#5) {} ; %
\node[coordinate] (c) at (#6) {} ; %
\node[coordinate] (a') at ($(b)!#1!(a)$) {} ; %
\node[coordinate] (c') at ($(b)!#2!(c)$) {} ; %
\node[coordinate] (d) at (barycentric cs:a'=1,c'=1,b=-1) {} ; %
\node[coordinate] (aup) at ($(a')!#3!(d)$) {};
\node[coordinate] (cup) at ($(c')!#3!(d)$) {};
\path[#7] (aup) -- (d) -- (cup) ; %
}
\newcommand{\pullback}[5][\pbkdefault]{\pullbackk[#1][#1]{#2}{#3}{#4}{#5}}
\newcommand{\pbk}[4][\pbkdefault]{%
\pullbackk[#1][#1]{#2}{#3}{#4}{draw,-} }
\newcommand{\stdpbk}{\pbk{m-2-1}{m-1-1}{m-1-2}}
\newcommand{\stdpo}{\pbk{m-2-1}{m-2-2}{m-1-2}}
\newcommand{\onepbk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,densely dotted} }
\newcommand{\ptwpbk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,densely dotted} }
\newcommand{\wpbk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,dashed} }
\newcommand{\pbkk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{cell=0} }
\newcommand{\laxpbk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,->,cell=0} }
\newcommand{\oplaxpbk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,<-,cell=0} }
\newcommand{\poleftg}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,open triangle 45-} }
\newcommand{\porightg}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,open triangle 45 reversed-} }
\newcommand{\dpbk}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,-open triangle 45} }
\newcommand{\dpbkrev}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,-open triangle 45 reversed} }
\newcommand{\dpbkblack}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,-triangle 45} }
\newcommand{\dpbkblackrev}[4][\pbkdefault]{%
\pullback[#1]{#2}{#3}{#4}{draw,triangle 45-open triangle 45} }
\tikzset{shortenlr/.style 2 args={shorten <={#1 ex},shorten >={#2 ex}}}
\tikzset{
back/.style={densely dotted}
}
\tikzset{
fore/.style 2 args={preaction={draw={white},-,line width=4pt,shorten <=#1cm,shorten >=#2cm}},
fore/.default={0.2}{0.2}
}
\tikzset{
foretwo/.style={preaction={draw=white,-,line width=6pt}}
}
\tikzset{twocell/.style = {double equal sign distance,double,-implies,shorten <= .15cm,shorten >=.15cm,draw}}
\tikzset{
cell/.style = {double equal sign distance,double,-implies,shorten <= #1 cm,shorten >=#1 cm,draw}
}
\tikzset{celllr/.style 2 args = {double equal sign distance,double,-implies,shorten <= #1 ex,shorten >=#2 ex,draw}}
\tikzset{identity/.style = {double equal sign distance,double,-,draw}}
\tikzset{iso/.style = {label={[below=0em,sloped]{$\scriptstyle -$}},label={[above=-.2em,sloped]{$\scriptstyle \sim$}}}}
\tikzset{equi/.style = {label={[above=-.2em,sloped]{$\scriptstyle \sim$}}}}
\tikzset{isotwo/.style = postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\node[inner sep=2pt,outer sep=0,above=-.2em,sloped] {$\scriptstyle \sim$} ;}
}}}
\tikzset{isor/.style = {labelr={\iso}}}
\tikzset{isol/.style = {labell={\iso}}}
\tikzset{isod/.style = {labeld={\iso}}}
\tikzset{isobr/.style = {labelbr={\iso}}}
\tikzset{baseline= (current bounding box.center)}
\tikzset{foreground/.style = {draw=white,very thick,double=black}}
\tikzset{background/.style = {draw=white,-,line width=3pt}}
\tikzset{diagram/.style = {column sep=1.5cm,row sep=1cm,nodes={minimum width=1cm}}}
\tikzset{nuage/.style = {cloud,draw,minimum width=2cm,minimum height=.6cm,cloud puffs=30,anchor=center}}
\tikzset{into/.style = {right hook->}}
\tikzset{otni/.style = {<-left hook}}
\tikzset{linto/.style = {left hook->}}
\tikzset{leadsto/.style = {->,decorate,decoration={snake,amplitude={#1 pt}}}}
\tikzset{leadsto/.default = {1.5}}
\tikzset{otsdael/.style = {<-,decorate,decoration={snake,amplitude=1.5pt}}}
\tikzset{onto/.style = {->>}}
\tikzset{mono/.style = {>->}}
\tikzset{fib/.style = {-latex}}
\tikzset{dfib/.style = {-Latex[open]}}
\tikzset{generic/.style = {-Triangle[]}}
\tikzset{free/.style = {-Triangle[open]}}
\tikzset{final/.style = {-]}}
\tikzset{epi/.style = {->>}}
\tikzset{adj/.style 2 args={text height={#1},text depth={#2}}}
\tikzset{adj/.default={.1cm}{0cm}}
\tikzset{iff/.style = {double equal sign distance,double,-implies,draw,shorten <= #1cm}}
\tikzset{
mod/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\draw[-] (0pt,-2pt) -- (0pt,2pt);}
}}},
negate/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\node[transform shape] (tempnode) {$/$};}
}}},
mapsto/.style={|->},
otspam/.style={<-|},
pro/.style={postaction={
decorate,
decoration={
markings,
mark=at position #1 with {\draw[-,fill] (0pt,0pt) circle (1.5pt);}
}}},
pro/.default={.5},
glob/.style={postaction={
decorate,
decoration={
markings,
mark=at position .5 with {\draw[-,fill=white] (0pt,0pt) circle (1.5pt);}
}}}
}
\newcommand{\idto}{\mathbin{\tikz[baseline] \draw[identity] (0pt,.5ex) -- (3ex,.5ex);}}
\newcommand{\idot}{\mathbin{\tikz[baseline] \draw[identity] (3ex,.5ex) -- (0pt,.5ex);}}
\newcommand{\fibto}[1]{\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[fib,labelu={\scriptstyle #1}] (3ex,.5ex);}}
\newcommand{\fibot}[1]{\mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[fib,labelu={\scriptstyle #1}] (0pt,.5ex);}}
\newcommand{\dfibto}{
\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib] (3ex,.5ex);}}
\newcommand{\xdfibto}[1]{
\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib,labelu={\scriptstyle #1}] (3ex,.5ex);}}
\newcommand{\shortdfibto}{
\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib] (1.5ex,.5ex);}}
\newcommand{\shortdfibot}{
\mathbin{\tikz[baseline] \draw (1.5ex,.5ex) edge[dfib] (0,.5ex);}}
\newcommand{\xxto}[2]{\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[->,labelu={\scriptstyle #1},labelb={\scriptstyle #2}] (3ex,.5ex);}}
\newcommand{\dfibot}[1][]{
\mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[dfib,labelu={\scriptstyle #1}] (0pt,.5ex);}}
\newcommand{\modto}{\mathbin{\tikz[baseline] \draw[->,mod] (0pt,.5ex) -- (3ex,.5ex);}}
\newcommand{\proto}{\mathbin{\tikz[baseline] \draw[->,pro] (0pt,.5ex) -- (3ex,.5ex);}}
\newcommand{\shortproto}{\mathbin{\tikz[baseline] \draw[->,pro] (0pt,.5ex) -- (1.5ex,.5ex);}}
\newcommand{\finalto}{\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[final] (3ex,.5ex);}}
\newcommand{\xfinalto}[1]{\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[final,labelu={\scriptstyle #1}] (3ex,.5ex);}}
\newcommand{\finalot}{\mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[final] (0pt,.5ex);}}
\newcommand{\xfinalot}[1]{\mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[final,labelu={\scriptstyle #1}] (0pt,.5ex);}}
\newcommand{\shortinto}{\mathbin{\tikz[baseline] \draw[->,into] (0pt,.5ex) -- (2ex,.5ex);}}
\newcommand{\cellule}[3][]{ %
\path (#3) +(#2:-.4cm) [twocell,#1] -- +(#2:.4cm) ; %
}
\newcommand{\celluled}[2]{ %
\cellule[labell={#2}]{-90}{#1} %
}
\newcommand{\celluler}[2]{ %
\cellule[labelu={#2}]{0}{#1} %
}
\newcommand{\isopath}[2]{%
\path %
(#1) -- node[sloped] {$\iso$} (#2) ; %
}
\newcommand{\adjtemplate}[6][1]{%
\begin{tikzpicture}[baseline=(m-1-1.base)] %
\matrix (m) [diag={1}{#1},
column sep={#1 cm,between borders}]{ %
|[adj]| #2 %
\& #6 %
\& |[adj]| #3 \\
} ; %
\path[->] %
(m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) %
(m-1-3.base west) edge[labeld={#5},bend left=15] (m-1-1.base east) %
; %
\end{tikzpicture} %
}
\newcommand{\adj}[5][1]{\adjtemplate[#1]{#2}{#3}{#4}{#5}{\bot}}
\newcommand{\coadj}[5][1]{\adjtemplate[#1]{#2}{#3}{#4}{#5}{\top}}
\newcommand{\catequi}[5][1]{\adjtemplate[#1]{#2}{#3}{#4}{#5}{\simeq}}
\newcommand{\catiso}[5][1]{\adjtemplate[#1]{#2}{#3}{#4}{#5}{\cong}}
\newcommand{\adjunction}[4]{%
\path[->] %
(#1) edge[twou={#3},bend left=15] (#2) %
(#2) edge[twod={#4},bend left=15] (#1) %
; %
\path (u) -- node[pos=.5,sloped] {$\dashv$} (d) ; %
}
\newcommand{\ladjunction}[4]{%
\path[->] %
(#1) edge[twou={#3},bend left=15] (#2) %
(#2) edge[twod={#4},bend left=15] (#1) %
; %
\path (u) -- node[pos=.5,sloped] {$\vdash$} (d) ; %
}
\newcommand{\adjs}[8][1]{%
\begin{tikzpicture} %
\matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ %
|[adj]| #2 %
\& \bot %
\& |[adj]| #3
\& \bot %
\& |[adj]| #6 \\
} ; %
\path[->] %
(m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) %
(m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) %
(m-1-3.north east) edge[labelu={#7},bend left=15] (m-1-5.north west) %
(m-1-5.south west) edge[labeld={#8},bend left=15] (m-1-3.south east) %
; %
\end{tikzpicture} %
}
\newcommand{\coadjs}[8][1]{%
\begin{tikzpicture} %
\matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ %
|[adj]| #2 %
\& \top %
\& |[adj]| #3
\& \top %
\& |[adj]| #6 \\
} ; %
\path[->] %
(m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) %
(m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) %
(m-1-3.north east) edge[labelu={#7},bend left=15] (m-1-5.north west) %
(m-1-5.south west) edge[labeld={#8},bend left=15] (m-1-3.south east) %
; %
\end{tikzpicture} %
}
\newcommand{\retr}[5][1]{%
\begin{tikzpicture} %
\matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ %
|[anchor=east,text height=.1cm,text depth=-.1cm]| #2 %
\& |[anchor=center]| \triangleleft %
\& |[anchor=west,text height=.1cm,text depth=-.1cm]| #3 \\
} ; %
\path[->] %
(m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) %
(m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) %
; %
\end{tikzpicture} %
}
\newcommand{\doublecell}[9]{
\diag{%
|(X)| {#1} \& |(Y)| {#2} \\ %
|(U)| {#3} \& |(V)| {#4} %
}{%
(X) edge[labelu={#5}] (Y) %
edge[pro,twol={#6}] (U) %
(Y) edge[pro,twor={#7}] (V) %
(U) edge[labeld={#8}] (V) %
(l) edge[cell=.4,labelu={\scriptstyle #9}] (r) %
}
}
\newcommand{\doublecellpro}[9]{
\diag{%
|(X)| {#1} \& |(Y)| {#2} \\ %
|(U)| {#3} \& |(V)| {#4} %
}{%
(X) edge[labelu={#5}] (Y) %
edge[pro,twol={#6}] (U) %
(Y) edge[pro,twor={#7}] (V) %
(U) edge[labeld={#8}] (V) %
(l) edge[cell=.4,labelu={\scriptstyle #9}] (r) %
}
}
\newcommand{\vdoublecell}[9]{
\diag{%
|(X)| {#1} \& |(Y)| {#2} \\ %
|(U)| {#3} \& |(V)| {#4} %
}{%
(X) edge[twou={#5}] (Y) %
edge[labell={#6}] (U) %
(Y) edge[labelr={#7}] (V) %
(U) edge[twod={#8}] (V) %
(u) edge[cell=.4,labelr={\scriptstyle #9}] (d) %
}
}
\newcommand{\Vdots}{|[anchor=center,text height=.1cm]| \vdots}
\newcommand{\relate}[3]{\path (#1) -- node[anchor=mid] {$#2$} (#3) ;}
\newcommand{\justify}[4][0.5]{\path (#2) -- node[anchor=mid,pos=#1] {\tiny (#3)} (#4) ;}
\newcommand{\mkdots}[2]{ \path
(#1) --
node[pos=.4] {.}
node[pos=.5] {.}
node[pos=.6] {.}
(#2)
;}
\newcommand{\mkdotsshrink}[2]{ \path
(#1) --
node[pos=.3] {.}
node[pos=.5] {.}
node[pos=.7] {.}
(#2)
;}
\endinput