From fec2806d5835d3734cdb4bc815640c7092e12f6e Mon Sep 17 00:00:00 2001
From: saundersp <pierre.saundersgb@gmail.com>
Date: Wed, 12 Feb 2025 19:00:06 +0100
Subject: [PATCH] contents/category_theory.tex : Moved homomorphism before
 epimorphism and added monomorphism definition

---
 contents/category_theory.tex | 58 +++++++++++++++++++++---------------
 references/annexes.bib       |  4 +++
 2 files changed, 38 insertions(+), 24 deletions(-)

diff --git a/contents/category_theory.tex b/contents/category_theory.tex
index a1f8fe5..2b4abe7 100644
--- a/contents/category_theory.tex
+++ b/contents/category_theory.tex
@@ -16,6 +16,23 @@ A category $\Cat$ is a collection of objects and morphisms
 A morphism $f$ on a category $\Cat$ is a transformation between a domain and a codomain.
 \end{definition_sq}
 
+\langsubsection{Homomorphisme}{Homomorphism}
+
+Source : \citeannexes{wikipedia_homomorphism}
+
+\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}
+
 \langsubsection{Section et rétraction}{Section and retraction}
 
 let $\function{f}{X}{Y}$ and $\function{g}{Y}{X}$ such that $f \composes g = \identity_Y$
@@ -35,18 +52,29 @@ Right inverse of a morphism, is the dual of a retraction. A section that is also
 
 Left inverse of a morphism, is the dual of a section. A retraction that is also a monomorphism is an isomorphism
 
+\langsubsection{Monomorphisme}{Monomorphism} \label{definition:monomorphism}
+
+Source : \citeannexes{wikipedia_monomorphism}
+
+A monomorphism is a homomorphism that is injective \ref{definition:injective}, similarly, a morphism that is left-cancellable i.e.
+
+Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$, $f \composes g_1 = f \composes g_2 \implies g_1 = g_2$.
+
+\[\begin{tikzcd}
+	Z \arrow[r, "g_1", shift left=1ex] \arrow[r, "g_2", shift right=1ex] & X \arrow[r, "f"] & Y
+\end{tikzcd}\]
+
 \langsubsection{Epimorphisme}{Epimorphism} \label{definition:epimorphism}
-%TODO Complete section
 
-Source: \citeannexes{wikipedia_epimorphism}
+Source : \citeannexes{wikipedia_epimorphism}
 
-Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$
+An epimorphism is a homomorphism that is surjective \ref{definition:surjective}, similarly, a morphism that is right-cancellable i.e.
 
-An epimorphism is a morphism that is right-cancellable i.e. $g_1 \composes f = g_2 \composes f \implies g_1 = g_2$
+Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$, $g_1 \composes f = g_2 \composes f \implies g_1 = g_2$.
 
-\begin{tikzcd}
+\[\begin{tikzcd}
 	X \arrow[r, "f"] & Y \arrow[r, "g_1", shift left=1ex] \arrow[r, "g_2", shift right=1ex] & Z
-\end{tikzcd}
+\end{tikzcd}\]
 
 \langsubsection{Isomorphisme}{Isomorphism} \label{definition:isomorphism}
 %TODO Complete section
@@ -67,24 +95,6 @@ Isomorphism is a bijective \ref{definition:bijection} morphism.
 
 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
-
-\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}
 %TODO Complete section
 
diff --git a/references/annexes.bib b/references/annexes.bib
index e05d904..efcc13f 100644
--- a/references/annexes.bib
+++ b/references/annexes.bib
@@ -349,6 +349,10 @@
   title = {Epimorphism},
   url   = {https://en.wikipedia.org/wiki/Epimorphism}
 }
+@online{wikipedia_monomorphism,
+  title = {Monomorphism},
+  url   = {https://en.wikipedia.org/wiki/Monomorphism}
+}
 @online{wikipedia_section_category_theory,
   title = {Section (category theory)},
   url   = {https://en.wikipedia.org/wiki/Section\_(category_theory)}