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.



A morphism f:YXf : Y \to X 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

Y× XY× XYtlY×Y× XYYfX \cdots Y \times_X Y \times_X Y \stackrel{\tl}{\stackrel{\to}{\to}} Y \times Y \times_X Y \stackrel{\to}{\to} Y \stackrel{f}{\to} X

is a 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).

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. 6 below):


A morphism of ∞-groupoids f:XYf \colon X \to Y is an effective epimorphism precisely if it is a surjection on connected components, hence if

π 0(f):π 0(X)π 0(Y) \pi_0(f) \colon \pi_0(X) \to \pi_0(Y)

is a surjection of sets.


As a corollary of prop. 5 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 on connected components.

This appears as HTT, cor.


Section 7.7 of

Section 6.2.3 of

Revised on August 18, 2013 14:24:05 by Urs Schreiber (