Categories

In mathematics, a category is a collection of objects and arrows between them that can be composed and have identities. Specifically, a category C consists of a class Ob(C) whose elements are objects, and for each pair (A,B) of objects a set Hom_C(A,B) of morphisms from A to B. For any object A there is an identity morphism id_A in Hom(A,A). If f in Hom(A,B) and g in Hom(B,C) then the composite g∘f is in Hom(A,C). Composition is associative, and identities act as neutral elements: id_B∘f = f and g∘id_B = g.

Examples: The category Set has objects sets and morphisms functions; Grp has groups and group homomorphisms; Top has topological spaces and continuous maps; Pos is the category arising from partially ordered sets where there is a single morphism A→B whenever A≤B.

In category theory, a functor F from C to D assigns to each object an object F(A) and to each morphism f: A→B a morphism F(f): F(A)→F(B), preserving identities and composition. A natural transformation between functors is a family of morphisms, one for each object, compatible with morphisms.

Categories can be equivalent or isomorphic; equivalence relaxes isomorphism of categories to being full, faithful, and essentially surjective on objects. Duality by reversing arrows leads to the opposite category C^op.

Many constructions generalize familiar set-based notions: limits and colimits, with products and coproducts as basic examples, as well as pullbacks and pushouts. The Yoneda lemma provides a powerful representation of objects by their relationships to other objects, and it is central to many developments in the subject.