nLab regular epimorphism in an (infinity,1)-category

Definition

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

$$\cdots c_2 \stackrel{\to}{\stackrel{\to}{\to}} c_1 \stackrel{\to}{\to} c \stackrel{f}{\to} d$$

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 not be an epimorphism in an (∞,1)-category (i.e. a monomorphism in the opposite category).

Questions

• 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?