The generalization of the notion of effective epimorphism from category theory to (∞,1)-category theory.
See also at 1-epimorphism.
A morphism in an (∞,1)-category is an effective epimorphism if it has a Cech nerve, of which it is the (∞,1)-colimit; in other words the augmented simplicial diagram
is an colimiting diagram.
This appears below HTT, cor. 184.108.40.206 for a (∞,1)-semitopos, but seems to be a good definition more generally.
In an (∞,1)-topos the effective epis are the n-epimorphisms for sitting in the (n-epi, n-mono) factorization system for with the monomorphism in an (∞,1)-category, factoring every morphism through its 1-image.
This appears as (Lurie, prop. 220.127.116.11).
This is HTT, cor. 18.104.22.168.
See n-connected/n-truncated factorization system for more on this.
This appears as (Rezk, lemma 7.9).
Useful is also the following characterization:
This is (Lurie, prop. 22.214.171.124).
This is true generally in the internal logic of the -topos (i.e. in homotopy type theory, see at 1-epimorphism for more on this), but in ∞Grpd sSet it is also true externally (prop. 7 below):
A morphism of ∞-groupoids is an effective epimorphism precisely if it is a surjection on connected components, hence if
is a surjection of sets.
In a sheaf -topos
In the infinity-topos of infinity-sheaves on an -site (i.e. in a topological localization), one has the following characterization of morphisms which become effective epimorphisms after applying the associated sheaf functor.
Let be an -site and a morphism of presheaves in . The morphism is an effective epimorphism in if and only if is a local epimorphism, i.e.
is -covering, or in other words
is a -covering sieve for all morphisms , where is the Yoneda embedding. (Here denotes the associated sheaf functor.)
This is clear. See MO/177325/2503 by David Carchedi for the argument.
As a corollary of prop. 5 we have
(effective epis of -groupoids)
In ∞Grpd a morphism is an effective epimorphism precisely if it induces an epimorphism in Set on connected components.
This appears as HTT, cor. 126.96.36.199.
Section 7.7 of
Section 6.2.3 of