The generalization of the notion of effective epimorphism from category theory to (∞,1)-category theory.
See also at 1-epimorphism. However, beware of the red herring principle: effective epimorphisms in an -category need not be epimorphisms.
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. 126.96.36.199 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. 188.8.131.52).
This is HTT, cor. 184.108.40.206.
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. 220.127.116.11).
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).
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. 18.104.22.168.
If denotes the homotopy type of the circle, then the unique morphism is an effective epimorphism, by prop. 7, but it not an epimorphism, because the suspension of is the sphere , which is not contractible.
Section 7.7 of
Section 6.2.3 of