Morphism

Epimorphism

A epimorphism is a right-cancellable morphism (denoted with ↠). A morphism 𝑒 :𝑋 →𝑌 is epic iff for any 𝑍 ∈𝖢 and 𝑓,𝑔 :𝑌 →𝑍 cat

𝑓∘𝑒=𝑔∘𝑒⟹𝑓=𝑔

In \Set a function is a epic iff it is surjective iff (assuming the Axiom of Choice) it is right-invertible (i.e. split epic), but these are not equivalent in every concrete category, rather:

graph LR;
  right-invertible ==>|implies| surjective ==>|implies| epic

Properties

See the dual properties.

1. If 𝑓𝑔 is epic then 𝑓 is epic.

Proof of 1

Dualize the corresponding proofs for properties of monomorphisms.

---

tidy | en | SemBr