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 $(\mathrm{\infty },1)$-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.

$$x\stackrel{f}{\to }y\overrightarrow{\to }{y}_1\overrightarrow{\overrightarrow{\to }}{y}_2\cdots$$x \stackrel{f}{\to} y \stackrel{\to}{\to} y_1 \stackrel{\to}{\stackrel{\to}{\to}} y_2 \cdots