nLab groupoid objects in an (∞,1)-topos are effective

Contents

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

In higher generalization (categorification) of how all equivalence relations in a Grothendieck topos have effective quotients, so in an \infty -sheaf \infty -topos all “\infty-equivalence relations”, constituted by \infty -groupoid objects have effective \infty -quotients. Where the former statement is one of the Giraud axioms that characterize sheaf toposes, the latter is one of the Giraud-Rezk-Lurie axioms that characterize \infty -sheaf \infty -toposes.

Statement

In an \infty -topos H\mathbf{H}, let X :Δ opHX_\bullet \,\colon\, \Delta^{op} \xrightarrow{\phantom{---}} \mathbf{H} be a simplicial object and write

(1)X 0q|X |lim[n]Δ opX nH X_0 \xrightarrow{\;\; q \;\;} \left\vert X_\bullet \right\vert \,\coloneqq\, \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim} X_n \;\;\; \in \; \mathbf{H}

for the \infty-quotient coprojection, i.e. induced universal morphism from X 0X_0 to the \infty -colimit. Then:

Proposition

(groupoid objects in an \infty-topos are effective)
If X X_\bullet is a groupoid object in that it satisfies the groupoidal Segal conditions, then the \infty-quotient coprojection (1) is an effective epimorphism in that X X_\bullet equivalent to its Cech nerve:

[n]Δ op(X nX 0×|X |×|X |X 0nfactors) \underset{[n] \in \Delta^{op}}{\prod} \Big( X_n \;\simeq\; \underset{n \; \text{factors}} { \underbrace{ X_0 \underset{\left\vert X_\bullet\right\vert}{\times} \cdots \underset{\left\vert X_\bullet\right\vert}{\times} X_0 } } \Big)

(Lurie 2009, Def. 6.1.2.14, Prop. 6.1.0.6 (3.iv))

Morever, this correspondence extends to morphisms:

Proposition

(equivalence between groupoids and effective epimoirphisms)
In any \infty -topos H\mathbf{H} the correspondence of Prop. extends to an equivalence of \infty -categories

(2)Grpd(H)() 0lim()() × (H Δ[1]) eff Grpd(\mathbf{H}) \underoverset { \underset{ (-)_0 \to \underset{\longrightarrow}{\lim}(-) } {\longrightarrow} } { \overset{ (-)^{\times_\bullet} } {\longleftarrow} } {\sim} \big( \mathbf{H}^{\Delta[1]} \big)_{eff}

between the groupoid objects in H \mathbf{H} and the full sub- \infty -category of the arrow category on the effective epimorphisms.

(Lurie 2009, below Cor. 6.2.3.5)

Here the \infty -functors are the restriction of L:Func(Δ op,H)Func([1],H)L \colon Func(\Delta^{op}, \mathbf{H}) \to \Func([1], \mathbf{H}) which sends a simplicial object to the universal morphism from X 0X_0 to its \infty -colimit and R:Func([1],H)Func(Δ op,H)R \colon \Func([1], \mathbf{H}) \to Func(\Delta^{op}, \mathbf{H}) sends a morphism to its Cech nerve.

This follows since the correspondence in both directions is computed by taking a Kan extension to Func(Δ + op,H)Func(\Delta_+^{op}, \mathbf{H}) followed by a restriction, and this identifies both sides with the same full subcategory of Func(Δ + op,H)Func(\Delta_+^{op}, \mathbf{H}).

Remark

(interpretation in terms of \infty-stacks equipped with atlases)
In as far as every object 𝒳H\mathcal{X} \,\in\, \mathbf{H} in an \infty -topos may be thought of an an \infty -stack, an effective epimorphism X𝒳X \twoheadrightarrow \mathcal{X} is an atlas for this \infty -stack and its Cech nerve is the \infty-groupoid that presents the \infty-stack.

For example, if H=\mathbf{H} \,=\, DTopGrpd DTopGrpd_\infty or == SmthGrpd SmthGrpd_\infty and X𝒳X \twoheadrightarrow \mathcal{X} is such that the Cech nerve takes values in 0-truncated concrete objects (using that these are cohesive \infty -toposes), then this recovers the traditional terminology in the field of topological groupoids/topological stacks and diffeological groupoids/differentiable stacks.

From this perspective, Prop. says that all \infty -toposes verify the expected relation between internal groupoids and geometric stacks equipped with atlases. For example, in this traditional terminology, a “Morita morphism” or “Hilsum-Skandalis morphism” between internal groupoids X X_\bullet, Y Y_\bullet is a morphism between their associated \infty-stacks |X ||Y |\left\vert X_\bullet \right\vert \to \left\vert Y_\bullet\right\vert, and, conversely, Prop. says that a morphism of \infty-stacks equipped with compatible atlases becomes equivalently a morphism of presenting groupoids.

See p. 27 in Sati Schreiber 2020.

Literature

Last revised on November 2, 2021 at 06:09:09. See the history of this page for a list of all contributions to it.