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

Idea

A regular monomorphism in a category is a morphism that is the equalizer of some pair of morphisms. A regular monomorphism in an $(\infty,1)$-category is its analog in an (∞,1)-category theory.

Beware that this need not be a monomorphism in an (∞,1)-category.

Definition

Let $C$ be an (∞,1)-category. A morphism $f : x \to y$ in $C$ is a regular monomorphism if there exists a cosimplicial diagram $D : \Delta \to C$ with $D[0] = y$ such that $f$ is the (∞,1)-limit over this diagram.

$$x \stackrel{f}{\to} y \stackrel{\to}{\to} y_1 \stackrel{\to}{\stackrel{\to}{\to}} y_2 \cdots$$