From 5bdb4ca5c26b4d6a81e21ca1bb250c96e9e86608 Mon Sep 17 00:00:00 2001 From: saundersp Date: Tue, 3 Dec 2024 19:44:15 +0100 Subject: [PATCH] contents/category_theory.tex : Added some basics properties and definitions --- contents/category_theory.tex | 88 ++++++++++++++++++++++++++++++++++++ references/annexes.bib | 44 ++++++++++++++++++ 2 files changed, 132 insertions(+) diff --git a/contents/category_theory.tex b/contents/category_theory.tex index 870104f..353e872 100644 --- a/contents/category_theory.tex +++ b/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 diff --git a/references/annexes.bib b/references/annexes.bib index 968143b..e15fa80 100644 --- a/references/annexes.bib +++ b/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} +}