Equivalence of categories: Difference between revisions
(New page: {{functor property}} ==Definition== ===Symbol-free definition=== An '''equivalence of categories]] is a functor between two categories that satisfies the following thre...) |
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An '''equivalence of categories | An '''equivalence of categories''' is a [[functor]] between two [[category|categories]] that satisfies the following three conditions: | ||
* It is [[full functor|full]] | * It is [[full functor|full]]: It is surjective on the collection of morphisms between any two objects. | ||
* It is [[faithful functor|faithful]] | * It is [[faithful functor|faithful]]: It is injective on the collection of morphisms between any two objects. | ||
* it is [[essentially surjective functor|essentially surjective]]. | * it is [[essentially surjective functor|essentially surjective]]: Every object in the target category is isomorphic to the image under the functor of some object in the source category. | ||
Latest revision as of 01:15, 9 December 2008
This article defines a functor property: a property that can be evaluated to true/false given a functor between two categories.
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Definition
Symbol-free definition
An equivalence of categories is a functor between two categories that satisfies the following three conditions:
- It is full: It is surjective on the collection of morphisms between any two objects.
- It is faithful: It is injective on the collection of morphisms between any two objects.
- it is essentially surjective: Every object in the target category is isomorphic to the image under the functor of some object in the source category.