List of Types of Functions - Relation To Category Theory

Relation To Category Theory

Category Theory is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms. A category is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a set of morphisms. A partial (equiv. dependently typed) binary operation called composition is provided on morphisms, every object has one special morphism from it to itself called the identity on that object, and composition and identities are required to obey certain relations.

In a so-called concrete category, the objects are associated with mathematical structures like sets, magmas, groups, rings, topological spaces, vector spaces, metric spaces, partial orders, differentiable manifolds, uniform spaces, etc., and morphisms between two objects are associated with structure-preserving functions between them. In the examples above, these would be functions, magma homomorphisms, group homomorphisms, ring homomorphisms, continuous functions, linear transformations (or matrices), metric maps, monotonic functions, differentiable functions, and uniformly continuous functions, respectively.

As an algebraic theory, one of the advantages of category theory is to enable one to prove many general results with a minimum of assumptions. Many common notions from mathematics (e.g. surjective, injective, free object, basis, finite representation, isomorphism) are definable purely in category theoretic terms (cf. monomorphism, epimorphism).

Category theory has been suggested as a foundation for mathematics on par with set theory and type theory (cf. topos).

Allegory theory provides a generalization comparable to category theory for relations instead of functions.