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

contents/category_theory.tex

@@ -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
 

references/annexes.bib

@@ -333,3 +333,47 @@
   title = {Hyperbolic functions},
   url   = {https://en.wikipedia.org/wiki/Hyperbolic\_functions}
 }
+@online{wikipedia_homomorphism,
+  title = {Homomorphism},
+  url   = {https://en.wikipedia.org/wiki/Homomorphism}
+}
+@online{wikipedia_morphism,
+  title = {Morphism},
+  url   = {https://en.wikipedia.org/wiki/Morphism}
+}
+@online{wikipedia_linearity,
+  title = {Linearity},
+  url   = {https://en.wikipedia.org/wiki/Linearity}
+}
+@online{wikipedia_epimorphism,
+  title = {Epimorphism},
+  url   = {https://en.wikipedia.org/wiki/Epimorphism}
+}
+@online{wikipedia_section_category_theory,
+  title = {Section (category theory)},
+  url   = {https://en.wikipedia.org/wiki/Section\_(category_theory)}
+}
+@online{wikipedia_ordered_pair,
+  title = {Ordered pair},
+  url   = {https://en.wikipedia.org/wiki/Ordered\_pair}
+}
+@online{bibmaths_regle_alembert,
+  title = {Règle de d'Alembert},
+  url   = {https://www.bibsmath.net/dico/index.php?action=affiche\&quoi=./r/regledalembert.html}
+}
+@online{bibmaths_regle_cauchy,
+  title = {Règle de Cauchy},
+  url   = {https://www.bibmath.net/dico/index.php?action=affiche\&quoi=./r/reglecauchy.html}
+}
+@online{maths_adultes_series_numerique_1,
+  title = {Séries numériques 1/6 : Tous les résultats à connaître},
+  url   = {https://www.youtube.com/watch?v=Vs9tBn0rypw}
+}
+@online{bibmaths_transformation_critere_abel,
+  title = {Transformation et critère d'Abel},
+  url   = {https://www.bibmath.net/dico/index.php?action=affiche\&quoi=./a/abeltransfo.html}
+}
+@online{bibmaths_critere_dirichlet,
+  title = {Critère de Dirichlet},
+  url   = {https://www.bibmath.net/dico/index.php?action=affiche\&quoi=./d/dirichletcritere.html}
+}