Contents

Definition

A strong epimorphism in a category $C$ is an epimorphism which is left orthogonal to any monomorphism in $C$.

Properties

• The composition of strong epimorphisms is a strong epimorphism. If $f\circ g$ is a strong epimorphism, then $f$ is a strong epimorphism.

• If $C$ has equalizers, then any morphism which is left orthogonal to all monomorphisms must automatically be an epimorphism.

• Every regular epimorphism is strong. The converse is true if $C$ is regular.

• Every strong epimorphism is extremal. The converse is true if $C$ has pullbacks.

In higher category theory

A monomorphism in an (∞,1)-category is a (-1)-truncated morphism in an (∞,1)-category $C$.

Therefore it makes sense to define an strong epimorphism in an $(\infty,1)$-category to be a morphism that is part of the left half of an orthogonal factorization system in an (∞,1)-category whose right half is that of $(-1)$-truncated morphisms.

If $C$ is an (∞,1)-topos then it has an n-connected/n-truncated factorization system for all $n$. The $(-1)$-connected morphisms are also called effective epimorphisms. Therefore in an $(\infty,1)$-topos strong epimorphisms again coincide with effective epimorphisms.

References

Textbook accounts:

• Francis Borceux, Def. 4.3.5 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)