A number of sophisticated people tend to disparage category theory as
consistently as others disparage certain kinds of classical music. When obliged.
to speak of a category they do so in an apologetic tone, similar to the way some
sa,y, “It was a gift-I’ve never even played it” when a record of Chopin
Nocturnes is discovered in their possession. For this reason I add to the usual
prerequisite that the reader have a fair amount of mathematical sophistication,
the further prerequisite that he have no other kind.
Functors, categories, natural transformations, and duality were introduced
in the early 1940’s by Eilenberg and MacLane [ 10,11]. Originally, the purpose
of these notions was to provide a technique for clarifying certain coficepts,
such as that of natural isomorphism. Category theory as a field in itself lay
relatively dormant during the following ten years. Nevertheless some work was
done by MacLane [28, 291, who introduced the important idea of defining
kernels, cokernels, direct sums, etc. , in terms of universal mapping properties
rather than in terms of the elements of the objects involved. MacLane also
gave some insight into the nature of the duality principle, illustrating it with
the dual nature of the frees and the divisibles in the category of abelian groups
(the projectives and injectives, respectively, in that category). Then with the
writing of the book “Homological Algebra” by Cartan and Eilenberg [6], it
became apparent that most propositions concerning finite diagrams of
modules could be proved in a more general type of category and, moreover,
that the number of such propositions could be halved through the use of
duality. This led to a full-fledged investigation of abelian categories by
Buchsbaum [3] (therein called exact categories). Grothendieck’s paper [20]
soon followed, and in it were introduced the important notions of A.B.5
category and generators for a category. (The latter idea had been touched on
by MacLane [29] .) Since then the theory has flourished considerably, not only
in the direction of generalizing and simplifying much of the already known
theorems in homological algebra, but also in its own right, notably through
the imbedding theorems and their metatheoretic consequences.