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contents/category_theory.tex : Added some basics properties and definitions

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contents/category_theory.tex
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@@ -5,23 +5,111 @@ 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{Section et rétraction}{Section and retraction}
let $\function{f}{X}{Y}$ and $\function{g}{Y}{X}$ such that $f \composes g = \text{id}_Y$
$f$ 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 an monomorphism is an isomorphism
\langsubsection{Epimorphisme}{Epimorphism} \label{definition:epimorphism}
%TODO Complete section
Source: \citeannexes{wikipedia_epimorphism}
Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$
An epimorphism is a morphism that is right-cancellative i.e. $g_1 \composes f = g_2 \composes f \implies g_1 = g_2$
\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{Homomorphisme}{Homomorphism}
%TODO Complete section
Source: \citeannexes{wikipedia_homomorphism}
\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