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:cdf : 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:Δ opCc : \Delta^{op} \to C, such that ff is the colimiting cocone

c 2c 1cfd \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 eCe \in C the induced morphism f *:C(d,e)C(c,e)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)