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epimorphism in an (infinity,1)-category

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 ( , 1 ) (\infty,1) -category and, more generally, the concept of n-epimorphism.

Definition

Definition

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

𝒞(f,A):𝒞(Y,A)𝒞(X,A) \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 𝒳\mathcal{X} \,\in\, Grpd Grpd_\infty , if the terminal map 𝒳*\mathcal{X} \to \ast is an epimorphism in the sense of Def. then

  1. 𝒳\mathcal{X} is connected;

  2. π 1 ( 𝒳 ) \pi_1(\mathcal{X}) is perfect.

(Hoyois 2019, Lem. 3)

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

Remark

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

𝒳Map(BG*,𝒳)Map(BG,𝒳) \mathcal{X} \xrightarrow{Map(B G \to \ast, \mathcal{X})} Map(B G, \mathcal{X})

are always fully faithful, for all 𝒳Grpd 1\mathcal{X} \in Grpd_1, without further conditions on GG, notably so for non-trivial abelian groups GG. Explicitly, for the case 𝒳BK\mathcal{X} \simeq B K (to which the general situation reduces by extensivity), we have the coproduct decomposition

Map(BG,BK)cH Grp 1(G,K)BStab K(c)BKcH Grp 1(G,K)c1BStab K(c) 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 BGB G into a 1-groupoid has only a single component, corresponding to the point *BG\ast \to B G, but a pseudonatural transformation out of BGB G into a 2-groupoid (and higher) has in addition a component for each element of GG, which are not reflected on *BG\ast \to B G.

In generalization of Ex. :

Example

A morphism XYX\to Y between connected spaces is an epimorphism iff YY is formed via a Quillen plus construction from a perfect normal subgroup of the fundamental group π 1(X)\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 ( , 1 ) (\infty,1) -toposes is discussed in Hoyois 19.

Example

A morphism ABA \to B of E E_\infty -rings is an epimorphism iff BB is smashing over AA, i.e., if B ABB B \wedge_A B \approx B.

References