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Version 3 |
| Let $C$ and $D$ be two categories with functors $F\colon C \to D$ and $G\colon D \to C$. |
Let $C$ and $D$ be two categories with functors $F\colon C \to D$ and $G\colon D \to C$. |
| The functors $F$ and $G$ are an \textbf{equivalence of categories} if there are |
The functors $F$ and $G$ are an \textbf{equivalence of categories} if there are |
| natural isomorphisms $FG \cong \id_D$ and $GF \cong \id_C$. |
natural isomorphisms $FG \cong \id_D$ and $GF \cong \id_C$. |
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| Note, $F$ is left adjoint to $G$, and $G$ is right adjoint to $F$ as |
Note, $F$ is left adjoint to $G$, and $G$ is right adjoint to $F$ as |
| \[ |
\[ |
| \hom_D(F(c),d) \stackrel{G}{\longrightarrow} \hom_C(GF(c),G(d)) \longleftrightarrow \hom_C(c,G(d)). |
\hom_D(F(c),d) \stackrel{G}{\longrightarrow} \hom_C(GF(c),G(d)) \longleftrightarrow \hom_C(c,G(d)). |
| \] |
\] |
| And, $F$ is right adjoint to $G$, and $G$ is left adjoint to $F$ as |
And, $F$ is right adjoint to $G$, and $G$ is left adjoint to $F$ as |
| \[ |
\[ |
| \hom_C(G(d),c) \stackrel{F}{\longrightarrow} \hom_D(FG(d),F(c)) \longleftrightarrow \hom_D(d,F(c)). |
\hom_C(G(d),c) \stackrel{F}{\longrightarrow} \hom_D(FG(d),F(c)) \longleftrightarrow \hom_D(d,F(c)). |
| \] |
\] |
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In practical terms, two categories are equivalent if there is a \PMlinkname{fully}{FullFunctor} faithful functor $F\colon C \to D$, such that every object $d \in D$ is isomorphic to an object $F(c)$, for some $c \in C$.
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In practical terms, two categories are equivalent if there is a fully faithful functor $F\colon C \to D$, such that every object $d \in D$ is isomorphic to an object $F(c)$, for some $c \in C$.
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