n. 1. a structure consisting of a class of objects, denoted ObC, and a class of arrows (or morphisms), denoted ArC, such that disjoint classes of morphisms are associated with each pair of objects and the morphisms are associative with an identity under composition. Categories are useful as an abstract model for the study of structures and mappings that preserve these structures. For example, in this diagram of arrows,the objects are groups G and A, and the morphisms represented by the arrows are homomorphisms θ and φ, and the identity function Ψ. When the categories are small and the morphisms are functions, it is called a classical category. See also functor. 2. short for Baire category.