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natural equivalence (Definition)

A natural transformation between functors $ \tau : F\to G$ is called a natural equivalence (or a natural isomorphism) if there is a natural transformation $ \sigma : G\to F$ such that $ \tau\circ\sigma = {\rm id}_G$ and $ \sigma\circ\tau = {\rm id}_F$ where $ {\rm id}_F$ is the identity natural transformation on $ F$ (which for each object $ A$ gives the identity map $ F(A)\to F(A)$), and composition is defined in the obvious way (for each object compose the morphisms and it's easy to see that this results in a natural transformation).



"natural equivalence" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Other names:  naturally equivalent, natural isomorphism
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Cross-references: morphisms, obvious, composition, identity map, object, identity, functors, natural transformation
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This is version 2 of natural equivalence, born on 2002-02-10, modified 2003-07-21.
Object id is 1893, canonical name is NaturalEquivalence.
Accessed 5403 times total.

Classification:
AMS MSC: 18-00 (Category theory; homological algebra :: General reference works )
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