145 lines
4.5 KiB
TeX
145 lines
4.5 KiB
TeX
\langchapter{Théorie des Catégories}{Category theory}
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%TODO Complete chapter
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Category is a general theory of mathematical structures and their relations.
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\langsection{Définition}{Definition}
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\begin{definition_sq} \label{definition:category}
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A category $\Cat$ is a collection of objects and morphisms
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\end{definition_sq}
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\langsection{Morphismes}{Morphisms}
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%TODO Complete section
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\begin{definition_sq} \label{definition:morphism}
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A morphism $f$ on a category $\Cat$ is a transformation between a domain and a codomain.
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\end{definition_sq}
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\langsubsection{Homomorphisme}{Homomorphism}
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Source : \citeannexes{wikipedia_homomorphism}
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\begin{definition_sq} \label{definition:homomorphism}
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A homomorphism is a morphism between two categories that keeps algebraic structures. Let $(X, \star)$ and $(Y, \composes)$ two algebraic structures and let $\function{\phi}{X}{Y}$.
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$$\forall (x, y) \in X^2, \phi(x \star y) = \phi(x) \composes \phi(y)$$
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Similarly, such that the following diagram commutes :
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\[\begin{tikzcd}
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X \cartesianProduct X \arrow[r, "\phi \cartesianProduct \phi"] \arrow[d, "\star"] & Y \cartesianProduct Y \arrow[d, "\composes"] \\
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X \arrow[r, "\phi"] & Y
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\end{tikzcd}\]
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\end{definition_sq}
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\langsubsection{Section et rétraction}{Section and retraction}
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let $\function{f}{X}{Y}$ and $\function{g}{Y}{X}$ such that $f \composes g = \identity_Y$
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$f$ is a retraction of $g$ and $g$ is a section of $f$.
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\begin{tikzcd}
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Y \arrow[r, "g"] \arrow[rd, "1_Y", below] & X \arrow[d, "f"] \\
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& Y
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\end{tikzcd}
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\subsubsection{Section}
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Right inverse of a morphism, is the dual of a retraction. A section that is also an epimorphism is an isomorphism
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\langsubsubsection{Rétraction}{Retraction}
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Left inverse of a morphism, is the dual of a section. A retraction that is also a monomorphism is an isomorphism
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\langsubsection{Monomorphisme}{Monomorphism} \label{definition:monomorphism}
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Source : \citeannexes{wikipedia_monomorphism}
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A monomorphism is a homomorphism that is injective \ref{definition:injective}, similarly, a morphism that is left-cancellable i.e.
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Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$, $f \composes g_1 = f \composes g_2 \implies g_1 = g_2$.
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\[\begin{tikzcd}
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Z \arrow[r, "g_1", shift left=1ex] \arrow[r, "g_2", shift right=1ex] & X \arrow[r, "f"] & Y
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\end{tikzcd}\]
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\langsubsection{Epimorphisme}{Epimorphism} \label{definition:epimorphism}
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Source : \citeannexes{wikipedia_epimorphism}
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An epimorphism is a homomorphism that is surjective \ref{definition:surjective}, similarly, a morphism that is right-cancellable i.e.
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Let $\function{f}{X}{Y}$ and $\function{g_1,g_2}{Y}{Z}$, $g_1 \composes f = g_2 \composes f \implies g_1 = g_2$.
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\[\begin{tikzcd}
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X \arrow[r, "f"] & Y \arrow[r, "g_1", shift left=1ex] \arrow[r, "g_2", shift right=1ex] & Z
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\end{tikzcd}\]
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\langsubsection{Isomorphisme}{Isomorphism} \label{definition:isomorphism}
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%TODO Complete section
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%Source: \citeannexes{wikipedia_isomorphism}
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Isomorphism is a bijective \ref{definition:bijection} morphism.
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\langsubsection{Endomorphisme}{Endomorphism} \label{definition:endomorphism}
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%TODO Complete section
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%Source: \citeannexes{wikipedia_endomorphisme}
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\langsubsection{Automorphisme}{Automorphism} \label{definition:automorphism}
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%TODO Complete section
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%Source: \citeannexes{wikipedia_automorphism}
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An automorphism is a morphism that is both an isomorphism \ref{definition:isomorphism} and an endomorphism \ref{definition:endomorphism}.
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\langsubsection{Homeomorphisme}{Homeomorphism}
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%TODO Complete section
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%Source: \citeannexes{wikipedia_homeomorphism}
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\langsubsection{Diffeomorphisme}{Diffeomorphism}
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%TODO Complete section
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%Source: \citeannexes{wikipedia_diffeomorphism}
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% TODO See difference with an differentiable isomorphism endomorphism continuous map
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\langsubsection{Exemples}{Examples}
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\begin{tikzcd}
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T
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\arrow[drr, bend left, "x"]
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\arrow[ddr, bend right, "y"]
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\arrow[dr, dotted, "{(x,y)}" description] & & \\
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& X \times_Z Y \arrow[r, "p"] \arrow[d, "q"]
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& X \arrow[d, "f"] \\
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& Y \arrow[r, "g"]
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& Z
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\end{tikzcd}
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\begin{tikzcd}[column sep=tiny]
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& \pi_1(U_1) \ar[dr] \ar[drr, "j_1", bend left=20]
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&
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&[1.5em] \\
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\pi_1(U_1 \union U_2) \ar[ur, "i_1"] \ar[dr, "i_2"']
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&
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& \pi_1(U_1) \ast_{ \pi_1(U_1 \union U_2)} \pi_1(U_2) \ar[r, dashed, "\simeq"]
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& \pi_1(X) \\
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& \pi_1(U_2) \ar[ur]\ar[urr, "j_2"', bend right=20]
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&
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&
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\end{tikzcd}
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\section{Functors}
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%TODO Complete section
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\subsection{Monads}
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%TODO Complete subsection
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\langsection{Argument diagonal}{Diagonal argument}
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%TODO Complete section
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