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 not be an epimorphism in an (∞,1)-category (i.e. a monomorphism in the opposite category).

Questions

Last revised on May 14, 2018 at 05:46:55. See the history of this page for a list of all contributions to it.