equivalences in/of $(\infty,1)$-categories
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 $f : c \to d$ in an (∞,1)-category a regular epimorphism if it is the colimit of some simplicial diagram, i.e. if there exists a functor $c : \Delta^{op} \to C$, such that $f$ is the colimiting cocone
over this diagram.
Equivalently, this is a morphism such that for all objects $e \in C$ the induced morphism $f^* : C(d,e) \to C(c,e)$ is a regular monomorphism in an (∞,1)-category in the (∞,1)-category ∞Grpd.
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.
If $f$ has a Cech nerve and is a regular epimorphism above, does it follow that it is the colimit of its Cech nerve (that is, that it is an effective epimorphism in an (∞,1)-category)?
Is there a notion of a strict epimorphism in an (∞,1)-category?