Natural Transformations and Isomorphisms
A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.
If F and G are (covariant) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) in D such that for every morphism f : X → Y in C, we have ηY ∘ F(f) = G(f) ∘ ηX; this means that the following diagram is commutative:
The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that ηX is an isomorphism for every object X in C.
Read more about this topic: Category Theory
Famous quotes containing the word natural:
“Man is, then, only disguise, falsehood, hypocrisy—both in himself and in regard to others. He does not wish any one to tell him the truth; he avoids telling it to others; and all these dispositions, so removed from justice and reason, have a natural root in his heart.”
—Blaise Pascal (1623–1662)