PlanetMath: equivalence of categories

(more info)

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equivalence of categories

(Definition)

Let and be two categories with functors $F\colon C \to D$ and $G\colon D \to C$ . The functors and are an equivalence of categories if there are natural isomorphisms $FG \cong \mathrm{id}_D$ and $GF \cong \mathrm{id}_C$ .

Note, is left adjoint to , and is right adjoint to as

$\displaystyle \hom_D(F(c),d) \stackrel{G}{\longrightarrow} \hom_C(GF(c),G(d)) \longleftrightarrow \hom_C(c,G(d)).$

And,

is right adjoint to

, and

is left adjoint to

$\displaystyle \hom_C(G(d),c) \stackrel{F}{\longrightarrow} \hom_D(FG(d),F(c)) \longleftrightarrow \hom_D(d,F(c)).$

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 , for some $c \in C$ .

"equivalence of categories" is owned by mhale.

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