Category theory maps maths by arrows, not just objects
Invented mid-century by Samuel Eilenberg and Saunders Mac Lane for algebraic topology, category theory treats structures by how they relate. Objects sit at ends of morphisms—arrows you can compose when targets match sources. The same pattern unifies products, quotients, and duality across fields, and informs functional programming.
A category has objects and morphisms; each morphism has a source and a target, often drawn as an arrow. Composition is allowed when the first arrow’s target is the second’s source, and it behaves like function composition: associative, with an identity arrow on every object. Morphisms need not be functions—a monoid is a one-object category whose morphisms are the monoid’s elements. From the axioms there is exactly one identity per object.
The category Set is the textbook example: objects are sets, morphisms are functions, identity functions exist, and composition is associative. Relations among arrows are often shown in commutative diagrams. Morphisms earn special names: monomorphisms (left-cancellable), epimorphisms (right-cancellable), bimorphisms (both), isomorphisms (invertible), endomorphisms (same object), automorphisms (invertible endomorphisms), retractions arrows, and sections. Every retraction is epic; every section is monic.
Functors are structure-preserving maps between categories—morphisms in the category of small categories. A covariant functor F from C to D sends each object x to F(x) and each arrow f : x → y to F(f) : F(x) → F(y), preserving composition and identities. A contravariant functor reverses arrows, equivalently a covariant functor from the opposite category. Natural transformations relate two functors; when they are invertible one speaks of a natural isomorphism, expressing that two constructions yield “the same” result.
Because many constructions recur similarly in different subjects—quotient spaces, direct products, completions, dualities—categories give a common language. Computer science leans on the same ideas for functional programming and for semantics of languages. What began as topology bookkeeping became a portable grammar for mathematical structure.
Source: Category theory
