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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 (alias quotient morphism) 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. 6.2.3.5 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.
In an (∞,1)-semitopos, effective epimorphisms are stable under (∞,1)-pullback.
This appears as (Lurie, prop. 6.2.3.15).
For an (∞,1)-semitopos we have that is an effective epimorphism precisely if its (-1)-truncation is a terminal object in the over-(∞,1)-category .
This is HTT, cor. 6.2.3.5.
More generally,
The effective epimorphisms in any (∞,1)-topos are precisely the (-1)-connected morphisms, and form a factorization system together with the monomorphisms (the (-1)-truncated morphisms).
See n-connected/n-truncated factorization system for more on this.
For an (∞,1)-topos, a morphism in is effective epi precisely if the induced morphism on subobjects ((∞,1)-monos, they form actually a small set) by (∞,1)-pullback
is injective.
This appears as (Rezk, lemma 7.9) and (Lurie, prop. 6.2.3.10).
Useful is also the following characterization:
A morphism in an (∞,1)-topos is an effective epimorphism precisely if its 0-truncation is an effective epimorphism in the underlying 1-topos.
This is (Lurie, prop. 7.2.1.14).
In words this means that a map is an effective epimorphism if it induces an epimorphism on connected components.
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. below).
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. we have:
(effective epis of -groupoids)
In ∞Grpd a morphism is an effective epimorphism precisely if it induces an epimorphism in Set (a surjection) on connected components.
This appears as HTT, cor. 7.2.1.15.
If denotes the homotopy type of the circle, then the unique morphism is an effective epimorphism, by prop. , but it not an epimorphism, because the suspension of is the sphere , which is not contractible.
Charles Rezk, Section 7.7 of: Toposes and homotopy toposes (pdf)
Jacob Lurie, Section 6.2.3 of: Higher Topos Theory
Last revised on December 14, 2023 at 20:48:15. See the history of this page for a list of all contributions to it.