nLab
strong epimorphism

Contents

Definition

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

Properties

In higher category theory

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

Therefore it makes sense to define an strong epimorphism in an (,1)(\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)(-1)-truncated morphisms.

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

References

Textbook accounts: