equivalences in/of $(\infty,1)$-categories
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.
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.