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

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

One analog in (∞,1)-category theory of epimorphism in category theory. Beware that there are other variants such as effective epimorphism in an (infinity,1)-category and generally the concept of n-epimorphism.

## Definition

For $C$ an (∞,1)-category, a morphism $f \colon X \to Y$ in $\mathcal{C}$ is an epimorphism if for all objects $A \in \mathcal{C}$ the induced morphism on hom $\infty$-groupoids

$\mathcal{C}(f,A) \,\colon\, \mathcal{C}(Y,A) \xhookrightarrow{\;\;} \mathcal{C}(X,A)$

is a monomorphism in ∞Grpd.

## Examples

• A morphism $A \to B$ of E-infinity rings is an epimorphism iff $B$ is smashing over $A$, i.e., if $B \wedge_A B \approx B$.

• A morphism $X\to Y$ between connected spaces is an epimorphism iff $Y$ is formed via a Quillen plus construction from a perfect normal subgroup of the fundamental group $\pi_1(X)$.

More generally, a map of spaces is an epi iff all its fibers are acyclic spaces in the sense that their suspensions are contractible. This is discussed in Raptis 17.

• A generalization of this to epimorphisms and acyclic spaces in (infinity,1)-toposes is discussed in Hoyois 19.

## References

Last revised on November 22, 2021 at 03:03:28. See the history of this page for a list of all contributions to it.