Category

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 homC(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,[a] 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 1x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have 1b ∘ f = f = f ∘ 1a.
    From the 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.

The "processes" or "maps" between categories that preserve the structure given above are called functors.

Basic examples of categories:

• groups.
• category of sets
• powerset category
• topos
  Two important construction in a category $C$ are product in categories and coproduct in categories.

An important strategy in category theory is to use universal propertys.