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 $c:{\Delta }^{\mathrm{op}}\to C$, such that $f$ is the colimiting cocone

$$\cdots c_2\overrightarrow{\overrightarrow{\to }}{c}_1\overrightarrow{\to }c\stackrel{f}{\to }d$$\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 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.

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?