nLab effective epimorphism in an (infinity,1)-category




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 (,1)(\infty, 1)-category need not be epimorphisms.



A morphism f:YXf : Y \to X 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

Y× XY× XYY× XYYfX \cdots Y \times_X Y \times_X Y \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Y \times_X Y \stackrel{\longrightarrow}{\longrightarrow} Y \stackrel{f}{\longrightarrow} X

is an colimiting diagram.

This appears below HTT, cor. for CC a (∞,1)-semitopos, but seems to be a good definition more generally.



In an (∞,1)-topos the effective epis are the n-epimorphisms for n=1n = 1 sitting in the (n-epi, n-mono) factorization system for n=1n = 1 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.



For CC an (∞,1)-semitopos we have that f:XYf : X \to Y is an effective epimorphism precisely if its (-1)-truncation is a terminal object in the over-(∞,1)-category C/YC/Y.

This is HTT, cor.

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 CC an (∞,1)-topos, a morphism f:XYf : X \to Y in CC is effective epi precisely if the induced morphism on subobjects ((∞,1)-monos, they form actually a small set) by (∞,1)-pullback

f *:Sub(Y)Sub(X) f^* : Sub(Y) \to Sub(X)

is injective.

This appears as (Rezk, lemma 7.9) and (Lurie, prop.

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.


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 (,1)(\infty,1)-topos (i.e. in homotopy type theory, see at 1-epimorphism for more on this), but in ∞Grpd L whe\simeq L_{whe} sSet it is also true externally (prop. below).

In a sheaf \infty-topos

In the infinity-topos of infinity-sheaves on an \infty-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 (C,t)(C,t) be an \infty-site and f:FGf : F \to G a morphism of presheaves in P(C)P(C). The morphism a t(f)a_t(f) is an effective epimorphism in Shv t(C)Shv_t(C) if and only if ff is a local epimorphism, i.e.

colimCˇ(f)G \colim \check{C}(f) \longrightarrow G

is tt-covering, or in other words

colimCˇ(f)× Gh(X)h(X) \colim \check{C}(f) \times_G h(X) \longrightarrow h(X)

is a tt-covering sieve for all morphisms h(X)Gh(X) \to G, where hh is the Yoneda embedding. (Here a ta_t denotes the associated sheaf functor.)

This is clear. See MO/177325/2503 by David Carchedi for the argument.


In Grpd\infty Grpd

As a corollary of prop. we have:


(effective epis of \infty-groupoids)

In C=C = ∞Grpd a morphism f:YXf : Y \to X is an effective epimorphism precisely if it induces an epimorphism π 0f:π 0Yπ 0X\pi_0 f : \pi_0 Y \to \pi_0 X in Set (a surjection) on connected components.

This appears as HTT, cor.


If S 1=****S^1 = \ast \underset{\ast \coprod \ast}{\coprod} \ast denotes the homotopy type of the circle, then the unique morphism S 1Δ 0S^1 \to \Delta^0 is an effective epimorphism, by prop. , but it not an epimorphism, because the suspension of S 1S^1 is the sphere S 2S^2, which is not contractible.


Last revised on December 14, 2023 at 20:48:15. See the history of this page for a list of all contributions to it.