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equivalence of categories (Definition)

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 equivalence of categories if there are natural isomorphisms $ FG \cong \mathrm{id}_D$ and $ GF \cong \mathrm{id}_C$.

Note, $ F$ is left adjoint to $ G$, and $ G$ is right adjoint to $ F$ as

$\displaystyle \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
$\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 $ F(c)$, for some $ c \in C$.



"equivalence of categories" is owned by mhale.
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See Also: essentially surjective

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Cross-references: isomorphic, object, faithful functor, equivalent, terms, left adjoint, natural isomorphisms, functors, categories
There are 13 references to this entry.

This is version 4 of equivalence of categories, born on 2003-02-26, modified 2005-05-22.
Object id is 4067, canonical name is EquivalenceOfCategories.
Accessed 3555 times total.

Classification:
AMS MSC18A40 (Category theory; homological algebra :: General theory of categories and functors :: Adjoint functors )

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