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effective epimorphism in an (infinity,1)-category

Contents

Idea

The generalization of the notion of effective epimorphism from category theory to (∞,1)-category theory.

See also at 1-epimorphism.

Definition

Definition

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{\longrightarrow}{\longrightarrow}} Y \times Y \times_X Y \stackrel{\longrightarrow}{\longrightarrow} Y \stackrel{f}{\longrightarrow} X

is an colimiting diagram.

This appears below HTT, cor. 6.2.3.5 for CC a (∞,1)-semitopos, but seems to be a good definition more generally.

Properties

Factorization

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.

Stability

Proposition

In an (∞,1)-semitopos, effective epimorphisms are stable under (∞,1)-pullback.

This appears as (Lurie, prop. 6.2.3.15).

Characterization

Proposition

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. 6.2.3.5.

More generally,

Proposition

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.

Proposition

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:

Proposition

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).

Remark

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. 7 below):

Example

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.

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.

Proposition

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.

Examples

As a corollary of prop. 5 we have

Proposition

(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. 7.2.1.15.

References

Section 7.7 of

Section 6.2.3 of

Revised on November 20, 2014 17:54:08 by Adeel Khan (132.252.63.48)