Isomorphism

An isomorphism is a fully invertible morphism, i.e. in a category $𝖢$ , $𝑓 \in 𝖢 (𝑋, 𝑌)$ is an isomorphism iff there exists $𝑓 - 1 \in 𝖢 (𝑌, 𝑋)$ such that #m/def/cat

i d 𝑋 = 𝑓 - 1 \circ 𝑓 i d 𝑌 = 𝑓 \circ 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.

#state/tidy | #lang/en | #SemBr