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:
“This seems a long while ago, and yet it happened since Milton wrote his Paradise Lost. But its antiquity is not the less great for that, for we do not regulate our historical time by the English standard, nor did the English by the Roman, nor the Roman by the Greek.... From this September afternoon, and from between these now cultivated shores, those times seemed more remote than the dark ages.”
—Henry David Thoreau (1817–1862)
“Or bid the soul of Orpheus sing
Such notes as, warbled to the string,
Drew iron tears down Pluto’s cheek,
And made Hell grant what love did seek;”
—John Milton (1608–1674)