category theory a mathematical theory that studies the universal properties of structures via their relationships with one another. A category C consists of two collections Obc and Morc, the objects and the morphisms of C, satisfying the following conditions: (i) for each pair (a, b) of objects there is associated a collection Morc (a, b) of morphisms such that each member of Morc belongs to one of these collections; (ii) for each object a of Obc, there is a morphism ida, called the identity on a; (iii) a composition law associating with each morphism f: a P b and each morphism g: b P c a morphism gf:a P c, called the composite of f and g; (iv) for morphisms f: a P b, g: b P c, and h: c P d, the equation h(gf) % (hg)f holds; (v) for any morphism f: a P b, we have idbf % f and fida % f. Sets with specific structures together with a collection of mappings preserving these structures are categories. Examples: (1) sets with functions between them; (2) groups with group homomorphisms; (3) topological spaces with continuous functions; (4) sets with surjections instead of arbitrary maps constitute a different category. But a category need not be composed of sets and set-theoretical maps. Examples: (5) a collection of propositions linked by the relation of logical entailment is a category and so is any preordered set; (6) a monoid taken as the unique object and its elements as the morphisms is a category. The properties of an object of a category are determined by the morphisms that are coming out of and going in this object. Objects with a universal property occupy a key position. Thus, a terminal object a is characterized by the following universal property: for any object b there is a unique morphism from b to a. A singleton set is a terminal object in the category of sets. The Cartesian product of sets, the product of groups, and the conjunction of propositions are all terminal objects in appropriate categories. Thus category theory unifies concepts and sheds a new light on the notion of universality. See also PHILOSOPHY OF MATHEMATIC. J.-P.M.