contents/category_theory.tex : Added homomorphism definition and useful macros

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}

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}