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

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saundersp
2025-02-09 22:10:22 +01:00
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contents/category_theory.tex
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@ -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}