Editing Natural transformation (section)

== Historical notes ==

[[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."<ref>{{harv|Mac Lane|1998|loc=§I.4}}</ref> Just as the study of [[group (mathematics)|groups]] is not complete without a study of [[group homomorphism|homomorphisms]], so the study of categories is not complete without the study of [[functor]]s. 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 (mathematics)|homology]]. Different ways of constructing homology could be shown to coincide: for example in the case of a [[simplicial complex]] the groups defined directly would be isomorphic to those of the singular theory.  What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.