CatDat

Implication Details

Assumptions: balanced, regular

Conclusions: epi-regular

Proof: Given any epimorphism $f : X \twoheadrightarrow Y$ in a regular category, we have the factorization into a regular epimorphism $X \twoheadrightarrow \im(f)$ followed by a monomorphism $\im(f) \hookrightarrow Y$. Because the composition is an epimorphism, the monomorphism $\im(f) \hookrightarrow Y$ must also be an epimorphism, and therefore an isomorphism. It follows that $f$ is in fact a regular epimorphism.