Functors and natural transformations

Eilenberg and MacLane invented (discovered) category theory in the early 1940s. They were working on Čech cohomology and wanted to separate the routine manipulations from those with more specific content. It turned out that category theory is good at that. Hence its other name abstract nonsense which is not always used with affection. Another part of their motivation was to try to explain why certain ‘natural’ constructions are natural, and other constructions are not. Such ‘natural’ constructions are now called natural transformations, a term that was used informally at the time but now has a precise definition. They observed that a natural transformation passes between two gadgets. These had to be made precise, and are now called functors. In turn each functor passes between two gadgets, which are now called categories. In other words, categories were invented to support functors, and these were invented to support natural transformations. But why the somewhat curious terminology? This is explained on pages 29 and 30 of Mac Lane (1998). … the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap … That, of course, is the bowdlerized version. Most of the basic notions were set up in Eilenberg and MacLane (1945) and that paper is still worth reading.