Natural transformation

In category theory a natural transformation is a process of transforming one functor into another in a way that respects the internal structure (the composition of morphisms) of the categories involved. For the precise definition and examples, see the article on category theory.

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. This much is obvious to any experienced mathematician. But the reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly, and those of the singular theory, would be isomorphic. But that in itself stated much less than the existence of a natural transformation of the corresponding homology functors.