This paper is a formulation of my personal opinion of the historical development and the present prospects of category theory.
Mathematics Subject Classification (1991). 18.02.
The study of categories is a natural and perhaps inevitable aspect of the 20th century mathematical emphasis on axiomatic and abstract methods -especially such as those methods when involved in abstract algebra and in functional analysis.
The origin of axiomatics lies in antiquity, but its modem prominence in mathematics was signaled by David Hilbert's famous book Grundlagen der Geometric. This book, first published in 1899, gave a careful and complete axiomatics for euclidean geometry, providing axioms with a unique model, to wit, the standard euclidean plane. But soon axioms were also used to describe selected common properties -those of whole families of mathematical objects, such as groups, rings, or topological spaces.
Emmy Noether, in G6ttingen as assistant to Hilbert, led this development in algebra. Her papers and her students emphasized the way in which a conceptual approach via axioms could clarify results and feature essential aspects of a subject -as in ring theory and in the remarkably perspicuous presentation of Galois theory provided, through her influence, in van der Waerden's Moderne Algebra and in Emil Artin's subsequent formulations (1944). Here, the central idea was that the Galois group consists of automorphisms of the field at issue. These developments showed how careful selection and use of concepts decisively influenced our understanding.
Categories constituted another natural step in this process of conceptual clarification. Dedekind, about 1900, had discovered lattices (which he called Dualgruppen). They made possible clear formulations of those aspects of structure involving inclusion of subobjects. The German algebraist H. Brandt, in his work to extend the Gauss composition of quadratic forms to biquadratic forms, found that these forms would not always compose; he was thus led to define groupoids of such forms (a groupoid is a category with all arrows isomorphisms). Curi-