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

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

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

## Definition

###### Definition

For $\mathcal{C}$ an (∞,1)-category, a 1-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

###### Example

(terminal epimorphisms of $\infty$-groupoids)
For $\mathcal{X} \,\in\,$ $Grpd_\infty$, if the terminal map $\mathcal{X} \to \ast$ is an epimorphism in the sense of Def. then

1. $\mathcal{X}$ is connected;

In particular, this means that in order for the delooping groupoid $\mathbf{B}G$ of a discrete group $G$ (i.e. the Eilenberg-MacLane space $K(G,1)$) to be such that $\mathbf{B}G \to \ast$ is epi, the group $G$ must be perfect.

###### Remark

(terminal epimorphisms of 1-groupoids) In contrast to Ex. , in the (2,1)-category $Grpd_1$ of 1-groupoids the maps

$\mathcal{X} \xrightarrow{Map(B G \to \ast, \mathcal{X})} Map(B G, \mathcal{X})$

are always fully faithful, for all $\mathcal{X} \in Grpd_1$, without further conditions on $G$, notably so for non-trivial abelian groups $G$. Explicitly, for the case $\mathcal{X} \simeq B K$ (to which the general situation reduces by extensivity), we have the coproduct decomposition

$Map(B G, \, B K) \;\simeq\; \underset{ c \in H^1_{Grp}(G,K) }{\coprod} B Stab_K(c) \;\;\;\simeq\;\;\; B K \, \sqcup \, \underset{ { c \in H^1_{Grp}(G,K) } \atop { c \neq 1 } }{\coprod} B Stab_K(c)$

which plays a central role in discussion such as of inertia orbifolds, equivariant principal bundles, equivariant K-theory and other aspects of equivariant cohomology.

The point is that a natural transformation out of $B G$ into a 1-groupoid has only a single component, corresponding to the point $\ast \to B G$, but a pseudonatural transformation out of $B G$ into a 2-groupoid (and higher) has in addition a component for each element of $G$, which are not reflected on $\ast \to B G$.

In generalization of Ex. :

###### Example

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 2017.

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

###### Example

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