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." 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. 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 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.
Read more about this topic: Natural Transformation
Famous quotes containing the words historical and/or notes:
“It is hard to believe that England is so near as from your letters it appears; and that this identical piece of paper has lately come all the way from there hither, begrimed with the English dust which made you hesitate to use it; from England, which is only historical fairyland to me, to America, which I have put my spade into, and about which there is no doubt.”
—Henry David Thoreau (1817–1862)
“There are only two or three human stories, and they go on repeating themselves as fiercely as if they had never happened before; like the larks in this country, that have been singing the same five notes over for thousands of years.”
—Willa Cather (1873–1947)