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.

Definition

Definition

A morphism $f:Y\to X$ 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

$$\cdots Y{\times }_{X}Y{\times }_{X}Y\stackrel{tl}{\overrightarrow{\to }}Y\times Y{\times }_{X}Y\overrightarrow{\to }Y\stackrel{f}{\to }X$$\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. 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

$$f^{*}:\mathrm{Sub}(Y)\to \mathrm{Sub}(X)$$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 $(\mathrm{\infty },1)$-topos (i.e. in homotopy type theory, see at 1-epimorphism for more on this), but in ∞Grpd $\simeq L_{\mathrm{whe}}$ sSet it is also true externally (prop. 6 below):

Example

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

$${\pi }_0(f):{\pi }_0(X)\to {\pi }_0(Y)$$\pi_0(f) \colon \pi_0(X) \to \pi_0(Y)

is a surjection of sets.

Examples

As a corollary of prop. 5 we have

Proposition

(effective epis of $\mathrm{\infty }$-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 on connected components.

This appears as HTT, cor. 7.2.1.15.

Related concepts

• effective epimorphism

• effective epimorphism in an $(\mathrm{\infty },1)$-category

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

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

References

Section 7.7 of

• Charles Rezk, Toposes and homotopy toposes (pdf)

Section 6.2.3 of

• Jacob Lurie, Higher Topos Theory