From 8ca8f213f9f734114e084e75c77eafa1d33fda85 Mon Sep 17 00:00:00 2001 From: saundersp Date: Sun, 9 Feb 2025 22:10:22 +0100 Subject: [PATCH] contents/category_theory.tex : Added homomorphism definition and useful macros --- contents/category_theory.tex | 19 ++++++++++++++++--- packages/macros.sty | 6 ++++++ 2 files changed, 22 insertions(+), 3 deletions(-) diff --git a/contents/category_theory.tex b/contents/category_theory.tex index 353e872..a1f8fe5 100644 --- a/contents/category_theory.tex +++ b/contents/category_theory.tex @@ -18,7 +18,7 @@ A morphism $f$ on a category $\Cat$ is a transformation between a domain and a c \langsubsection{Section et rétraction}{Section and retraction} -let $\function{f}{X}{Y}$ and $\function{g}{Y}{X}$ such that $f \composes g = \text{id}_Y$ +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 $f$. @@ -33,7 +33,7 @@ Right inverse of a morphism, is the dual of a retraction. A section that is also \langsubsubsection{Rétraction}{Retraction} -Left inverse of a morphism, is the dual of a section. A retraction that is also an monomorphism is an isomorphism +Left inverse of a morphism, is the dual of a section. A retraction that is also a monomorphism is an isomorphism \langsubsection{Epimorphisme}{Epimorphism} \label{definition:epimorphism} %TODO Complete section @@ -42,7 +42,7 @@ 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. $g_1 \composes f = g_2 \composes f \implies g_1 = g_2$ +An epimorphism is a morphism that is right-cancellable 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 @@ -70,6 +70,19 @@ An automorphism is a morphism that is both an isomorphism \ref{definition:isomor \langsubsection{Homomorphisme}{Homomorphism} %TODO Complete section +\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} + Source: \citeannexes{wikipedia_homomorphism} \langsubsection{Homeomorphisme}{Homeomorphism} diff --git a/packages/macros.sty b/packages/macros.sty index 168bd3e..8905bcc 100644 --- a/packages/macros.sty +++ b/packages/macros.sty @@ -19,17 +19,23 @@ \newcommand{\C}{\mathbb{C}} % Complex numbers symbol \newcommand{\Cat}{\mathcal{C}} % Category \newcommand{\Set}{\mathbf{Set}} % Set category +\newcommand{\Grp}{\mathbf{Grp}} % Group category +\newcommand{\Ab}{\mathbf{Ab}} % Abelian category +\newcommand{\Top}{\mathbf{Top}} % Topological spaces category \newcommand{\K}{\mathbb{K}} % Corps \newcommand{\Hq}{\mathbb{H}} % Quaternions numbers symbol \newcommand{\Ot}{\mathbb{O}} % Octonions numbers symbol \newcommand{\Se}{\mathbb{S}} % Sedenions numbers symbol \newcommand{\Pn}{\mathbb{P}} % Sets of all the prime numbers \newcommand{\B}{\mathbf{B}} % Topological Ball +\newcommand{\Identity}{\text{Id}} % Identity +\newcommand{\identity}{\text{id}} % identity \newcommand{\false}{F} % New symbol for false value \newcommand{\true}{V} % New symbol for true value \DeclareMathOperator{\Rel}{\mathcal{R}} % New symbol for binary relations \DeclareMathOperator{\composes}{\circ} % New symbol composing morphisms \DeclarePairedDelimiter{\abs}{|}{|} +\newcommand{\isomorphic}{\cong} % Isomorphism \DeclarePairedDelimiter{\card}{|}{|} \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}