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An extension of the concept of an isomorphism of categories brought about, first of all, by the presence of classes of isomorphic objects.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360502.png" /> are called equivalent if there are one-place covariant functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360504.png" /> such that the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360505.png" /> is naturally equivalent to the identity functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360506.png" /> and the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360507.png" /> to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360508.png" />; in other words, the categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e03605010.png" /> are equivalent if there are functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e03605011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e03605012.png" /> "almost"  inverse to one another. Two categories are equivalent if and only if their skeletons are isomorphic (see [[Skeleton of a category|Skeleton of a category]]).
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Two categories $\mathfrak{K}$ and $\mathfrak{L}$ are called equivalent if there are one-place covariant functors $F : \mathfrak{K} \rightarrow \mathfrak{L}$ and $G : \mathfrak{L} \rightarrow \mathfrak{K}$ 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 [[Skeleton of a category|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.
 
 
 
 
 
  
====Comments====
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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).
Cf. [[Triple|Triple]] for that notion, and the editorial comments to [[Category|Category]] for the notion of a Kleisli category of a triple.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR>
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</table>

Latest revision as of 07:37, 28 November 2017

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 $F : \mathfrak{K} \rightarrow \mathfrak{L}$ and $G : \mathfrak{L} \rightarrow \mathfrak{K}$ 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).

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
How to Cite This Entry:
Equivalence of categories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_of_categories&oldid=12063
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article