The composition of strong epimorphisms is a strong epimorphism. If is a strong epimorphism, then is a strong epimorphism.
If has equalizers, then any morphism which is left orthogonal to all monomorphisms must automatically be an epimorphism.
Therefore it makes sense to define an strong epimorphism in an -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 -truncated morphisms.
If is an (∞,1)-topos then it has an n-connected/n-truncated factorization system for all . The -connected morphisms are also called effective epimorphisms. Therefore in an -topos strong epimorphisms again coincide with effective epimorphisms.