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

Idea

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

Definition

Definition

A morphism $f : Y \to X$ 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

$$\cdots Y \times_X Y \times_X Y \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Y \times_X Y \stackrel{\longrightarrow}{\longrightarrow} Y \stackrel{f}{\longrightarrow} X$$

is an colimiting diagram.

This appears below HTT, cor. 6.2.3.5 for $C$ 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 $n = 1$ sitting in the (n-epi, n-mono) factorization system for $n = 1$ 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 $C$ an (∞,1)-semitopos we have that $f : X \to Y$ is an effective epimorphism precisely if its (-1)-truncation is a terminal object in the over-(∞,1)-category $C/Y$.

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

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

is injective.

This appears as (Rezk, lemma 7.9) and (Lurie, prop. 6.2.3.10).

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 $(\infty,1)$-topos (i.e. in homotopy type theory, see at 1-epimorphism for more on this), but in ∞Grpd $\simeq L_{whe}$ 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.

Proposition

Let $(C,t)$ be an $\infty$-site and $f : F \to G$ a morphism of presheaves in $P(C)$. The morphism $a_t(f)$ is an effective epimorphism in $Shv_t(C)$ if and only if $f$ is a local epimorphism, i.e.

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

is $t$-covering, or in other words

$$\colim \check{C}(f) \times_G h(X) \longrightarrow h(X)$$

is a $t$-covering sieve for all morphisms $h(X) \to G$, where $h$ is the Yoneda embedding. (Here $a_t$ denotes the associated sheaf functor.)

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

Examples

In $\infty Grpd$

As a corollary of prop. we have:

Proposition

(effective epis of $\infty$-groupoids)

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

This appears as HTT, cor. 7.2.1.15.

Example

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

Related concepts

• effective epimorphism

• effective epimorphism in homotopy type theory

• groupoid objects in an (∞,1)-topos are effective

• regular epimorphism in an (∞,1)-category

• n-connected object of an (∞,1)-category

• n-types cover

References

• Charles Rezk, Section 7.7 of: Toposes and homotopy toposes (pdf)

• Jacob Lurie, Section 6.2.3 of: Higher Topos Theory