\langchapter{Théorie des Catégories}{Category theory}
%TODO Complete chapter

Category is a general theory of mathematical structures and their relations.

\langsection{Définition}{Definition}

\begin{definition_sq} \label{definition:category}
A category $\Cat$ is a collection of objects and morphisms
\end{definition_sq}

\langsection{Morphismes}{Morphisms}
%TODO Complete section

\begin{definition_sq} \label{definition:morphism}
A morphism $f$ on a category $\Cat$ is a transformation between a domain and a codomain.
\end{definition_sq}

\langsubsection{Homomorphisme}{Homomorphism}

Source : \citeannexes{wikipedia_homomorphism}

\begin{definition_sq} \label{definition:homomorphism}
	A homomorphism is a morphism between two categories that keeps algebraic structures. Let $(X, \star)$ and $(Y, \composes)$ two algebraic structures and let $\function{\phi}{X}{Y}$.

	$$\forall (x, y) \in X^2, \phi(x \star y) = \phi(x) \composes \phi(y)$$

	Similarly, such that the following diagram commutes :

	\[\begin{tikzcd}
		X \cartesianProduct X \arrow[r, "\phi \cartesianProduct \phi"] \arrow[d, "\star"] & Y \cartesianProduct Y \arrow[d, "\composes"] \\
		X \arrow[r, "\phi"] & Y
	\end{tikzcd}\]
\end{definition_sq}

\langsubsection{Section et rétraction}{Section and retraction}

let $\function{f}{X}{Y}$ and $\function{g}{Y}{X}$ such that $f \composes g = \identity_Y$

$f$ is a retraction of $g$ and $g$ is a section of $f$.

\begin{tikzcd}
	Y \arrow[r, "g"] \arrow[rd, "1_Y", below] & X \arrow[d, "f"] \\
		& Y
\end{tikzcd}

\subsubsection{Section}

Right inverse of a morphism, is the dual of a retraction. A section that is also an epimorphism is an isomorphism

\langsubsubsection{Rétraction}{Retraction}

Left inverse of a morphism, is the dual of a section. A retraction that is also a monomorphism is an isomorphism

\langsubsection{Monomorphisme}{Monomorphism} \label{definition:monomorphism}

Source : \citeannexes{wikipedia_monomorphism}

A monomorphism is a homomorphism that is injective \ref{definition:injective}, similarly, a morphism that is left-cancellable i.e.

Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$, $f \composes g_1 = f \composes g_2 \implies g_1 = g_2$.

\[\begin{tikzcd}
	Z \arrow[r, "g_1", shift left=1ex] \arrow[r, "g_2", shift right=1ex] & X \arrow[r, "f"] & Y
\end{tikzcd}\]

\langsubsection{Epimorphisme}{Epimorphism} \label{definition:epimorphism}

Source : \citeannexes{wikipedia_epimorphism}

An epimorphism is a homomorphism that is surjective \ref{definition:surjective}, similarly, a morphism that is right-cancellable i.e.

Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$, $g_1 \composes f = g_2 \composes f \implies g_1 = g_2$.

\[\begin{tikzcd}
	X \arrow[r, "f"] & Y \arrow[r, "g_1", shift left=1ex] \arrow[r, "g_2", shift right=1ex] & Z
\end{tikzcd}\]

\langsubsection{Isomorphisme}{Isomorphism} \label{definition:isomorphism}
%TODO Complete section

%Source: \citeannexes{wikipedia_isomorphism}

Isomorphism is a bijective \ref{definition:bijection} morphism.

\langsubsection{Endomorphisme}{Endomorphism} \label{definition:endomorphism}
%TODO Complete section

%Source: \citeannexes{wikipedia_endomorphisme}

\langsubsection{Automorphisme}{Automorphism} \label{definition:automorphism}
%TODO Complete section

%Source: \citeannexes{wikipedia_automorphism}

An automorphism is a morphism that is both an isomorphism \ref{definition:isomorphism} and an endomorphism \ref{definition:endomorphism}.

\langsubsection{Homeomorphisme}{Homeomorphism}
%TODO Complete section

%Source: \citeannexes{wikipedia_homeomorphism}

\langsubsection{Diffeomorphisme}{Diffeomorphism}
%TODO Complete section

%Source: \citeannexes{wikipedia_diffeomorphism}

% TODO See difference with an differentiable isomorphism endomorphism continuous map

\langsubsection{Exemples}{Examples}

\begin{tikzcd}
	T
	\arrow[drr, bend left, "x"]
	\arrow[ddr, bend right, "y"]
	\arrow[dr, dotted, "{(x,y)}" description] & & \\
		& X \times_Z Y \arrow[r, "p"] \arrow[d, "q"]
			& X \arrow[d, "f"] \\
		& Y \arrow[r, "g"]
			& Z
\end{tikzcd}

\begin{tikzcd}[column sep=tiny]
	& \pi_1(U_1) \ar[dr] \ar[drr, "j_1", bend left=20]
		&
			&[1.5em] \\
	\pi_1(U_1 \union U_2) \ar[ur, "i_1"] \ar[dr, "i_2"']
		&
			& \pi_1(U_1) \ast_{ \pi_1(U_1 \union U_2)} \pi_1(U_2) \ar[r, dashed, "\simeq"]
				& \pi_1(X) \\
	& \pi_1(U_2) \ar[ur]\ar[urr, "j_2"', bend right=20]
		&
			&
\end{tikzcd}

\section{Functors}
%TODO Complete section

\subsection{Monads}
%TODO Complete subsection

\langsection{Argument diagonal}{Diagonal argument}
%TODO Complete section