By comparing the notions of regular epimorphism and effective epimorphism with that of effective epimorphism in an (∞,1)-category, we propose to call a morphism in an (∞,1)-category a regular epimorphism if it is the colimit of some simplicial diagram, i.e. if there exists a functor , such that is the colimiting cocone
over this diagram.
Warning. Such a morphism may fail to satisfy some condition for being a plain epimorphism in an (∞,1)-category that you might think of; in particular it need not be a monomorphism in an (∞,1)-category. The idea is that there may not be a good notion of epimorphism in an (∞,1)-category apart from regular epimorphism.