Morphism

Isomorphism

An isomorphism is a fully invertible morphism, i.e. in a category 𝖢, 𝑓 ∈𝖢(𝑋,𝑌) is an isomorphism iff there exists 𝑓−1 ∈𝖢(𝑌,𝑋) such that cat

id𝑋=𝑓−1∘𝑓id𝑌=𝑓∘𝑓−1

It is important to note that the inverse must exist in the same category, and hence

graph LR;
  bijection["bijection (concrete)"]
  mopic["monic and epic"]
  isomorphism ==>|implies| bijection ==>|implies| mopic

Consider, for example, a bijective continuous map that are not homeomorphism.

---

tidy | en | SemBr