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

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

## 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 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

Created on September 12, 2011 20:59:19 by Mike Shulman (71.136.248.27)