Equivalence of categories

2020 Mathematics Subject Classification: Primary: 18A05 [MSN][ZBL]

An extension of the concept of an isomorphism of categories brought about, first of all, by the presence of classes of isomorphic objects.

Two categories $\mathfrak{K}$ and $\mathfrak{L}$ are called equivalent if there are one-place covariant functors $F:\mathfrak{K}\to \mathfrak{L}$ and $G:\mathfrak{L}\to \mathfrak{K}$ such that the product $FG$ is naturally equivalent to the identity functor ${\mathrm{Id}}_{\mathfrak{L}}$ and the product $GF$ to the functor ${\mathrm{Id}}_{\mathfrak{K}}$; in other words, the categories $\mathfrak{K}$ and $\mathfrak{L}$ are equivalent if there are functors $F$ and $G$ "almost" inverse to one another. Two categories are equivalent if and only if their skeletons are isomorphic.

Pontryagin's duality theorem establishes the equivalence of the category of Abelian groups and the category that is dual to that of topological Abelian groups; the category of Boolean algebras is equivalent to the category that is dual to that of Boolean spaces; the category of binary relations over the category of sets is equivalent to the Kleisli category for the triple defined by the functor of taking the set of subsets (cf. the editorial comments to Category for the notion of a Kleisli category of a triple).

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