Category Theory - Categories, Objects, and Morphisms

Categories, Objects, and Morphisms

A category C consists of the following three mathematical entities:

• A class ob(C), whose elements are called objects;
• A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a source object a and target object b.
  The expression f : a → b, would be verbally stated as "f is a morphism from a to b".
  The expression hom(a, b) — alternatively expressed as hom_C(a, b), mor(a, b), or C(a, b) — denotes the hom-class of all morphisms from a to b.
• A binary operation ∘, called composition of morphisms, such that for any three objects a, b, and c, we have hom(b, c) × hom(a, b) → hom(a, c). The composition of f : a → b and g : b → c is written as g ∘ f or gf, governed by two axioms:

• Associativity: If f : a → b, g : b → c and h : c → d then h ∘ (g ∘ f) = (h ∘ g) ∘ f, and
• Identity: For every object x, there exists a morphism 1_x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have 1_b ∘ f = f = f ∘ 1_a.

From these axioms, it can be proved that there is exactly one identity morphism for every object. Some authors deviate from the definition just given by identifying each object with its identity morphism.

Relations among morphisms (such as fg = h) are often depicted using commutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms.