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