Main Notions of the Category Theory

Abstract

A category C consists of the following data:

• a) A class ObC whose elements are called objects of C.

• b) A collection of sets Hom(X, Y), one for each ordered pair of objects X, Y ∈ ObC, whose elements are called morphisms (from X to Y); they are denoted by ϕ: X → Y.

• c) A collection of mappings

  $$ Hom(X,Y) \times Hom(Y,Z) \to Hom(X,Z), $$

  one for each ordered triple of objects X,Y,Z ∈ ObC. Any mapping in this collection associates with a pair ϕ: X → Y, ψ: Y → Z a morphism from X to Z, denoted by ψ ◦ ϕ or ψϕ: X → Z, and called the composition or product of ϕ and ψ.