nLab epimorphism in an (infinity,1)-category

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;

2. $\pi_1(\mathcal{X})$ is perfect.

(Hoyois 2019, Lem. 3)

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 $B \wedge_A B \approx B$.

Related concepts

• Quillen plus construction

References

• George Raptis, Some characterizations of acyclic maps, Journal of Homotopy and Related Structures volume 14, pages 773–785 (2019) 2017 (arxiv:1711.08898, doi:10.1007/s40062-019-00231-6)

• Marc Hoyois, On Quillen’s plus construction, 2019 (pdf, pdf)

• Oren Ben-Bassat, D. Mukherjee, Analytification, localization and homotopy epimorphisms, Bulletin des Sciences Mathématiques 176 (2022) 103129 arXiv:2111.04184 doi