natural equivalenceNaturalEquivalence2002-02-10 15:30:572008-10-25 19:25:02Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $F,G: \mathcal{C}\to \mathcal{D}$ be a pair of functors from the category $\mathcal{C}$ to the category $\mathcal{D}$. A natural transformation between functors $\tau : F\to G$ is called a {\em natural equivalence} (or a {\em natural isomorphism}) if there is a natural transformation $\sigma : G\to F$ such that $\tau\bullet \sigma = {\rm id}_G$ and $\sigma\bullet \tau = {\rm id}_F$ where ${\rm id}_F$ is the identity natural transformation on $F$, and composition $\bullet$ is the usual (vertical) composition on natural transformations.
Equivalently, one can define a natural equivalence from functors $F$ to $G$ to be a natural transformation $\tau$ such that for each object $A$ in $\mathcal{C}$, the morphism $\tau_A : F(A)\to G(A)$ is an isomorphism in $\mathcal{D}$.