Functors

Functors - Definition

Let 𝐶 and 𝐷 be categories. A functor 𝐹 from 𝐶 to 𝐷 is a mapping that:

• associates each object 𝑋 in 𝐶 to an object 𝐹(𝑋) in 𝐷
• associates each morphism 𝑓 : 𝑋 → 𝑌 in 𝐶 to a morphism 𝐹(𝑓) : 𝐹(𝑋) → 𝐹(𝑌) in 𝐷 such that functor 𝐹 preserves the following conditions:
  • identity morphisms - 𝐹(𝑖𝑑_𝑋) = 𝑖𝑑_𝐹(𝑋) for every object 𝑋 in 𝐶
  • composition of morphisms - 𝐹(𝑔∘𝑓) = 𝐹(𝑔)∘𝐹(𝑓) for all morphisms 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑍 in 𝐶