nLab n-epimorphism

Definition

In an (∞,1)-topos a morphism $f \colon X \to Y$ is an $n$-epimorphism for $n \in \mathbb{N}$ equivalently if

• in its n-image factorization

  $$f \colon X \to im_n(f) \to Y$$

  the second morphism is an equivalence in an (∞,1)-category;

• it is an (n-2)-connected object in the slice (∞,1)-topos $\mathbf{H}_{/Y}$.

Properties

The $n$-epimorphisms in an (∞,1)-topos are the left half of the ((n-2)-epi, (n-2)-mono) factorization system which factors every morphism through its n-image.

Examples

• The $\infty$-epimorphisms are precisely the equivalences.

• The 1-epimorphism are the effective epimorphisms.

• Every morphism is a $0$-epimorphism.

• The 1-epimorphisms between 0-truncated objects are precisely the ordinary epimorphisms in the underlying 1-topos.

Related concepts

• Postnikov system

• n-types cover

References

• Jacob Lurie, Higher Topos Theory

Disucssion in homotopy type theory is in

• Univalent Foundations Project, section 7.6 of Homotopy Type Theory – Univalent Foundations of Mathematics