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

Context

$(\infty,1)$-Topos Theory

(∞,1)-topos theory

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 $\mathbf{H}$, let $X_\bullet \,\colon\, \Delta^{op} \xrightarrow{\phantom{---}} \mathbf{H}$ be a simplicial object and write

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

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

Proposition

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

$\underset{[n] \in \Delta^{op}}{\prod} \Big( X_n \;\simeq\; \underset{n \; \text{factors}} { \underbrace{ X_0 \underset{\left\vert X_\bullet\right\vert}{\times} \cdots \underset{\left\vert X_\bullet\right\vert}{\times} X_0 } } \Big)$

(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 $\mathbf{H}$ the correspondence of Prop. extends to an equivalence of $\infty$-categories

(2)$Grpd(\mathbf{H}) \underoverset { \underset{ (-)_0 \to \underset{\longrightarrow}{\lim}(-) } {\longrightarrow} } { \overset{ (-)^{\times_\bullet} } {\longleftarrow} } {\sim} \big( \mathbf{H}^{\Delta[1]} \big)_{eff}$
(Lurie 2009, below Cor. 6.2.3.5)

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

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

Remark

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

For example, if $\mathbf{H} \,=\,$ $DTopGrpd_\infty$ or $=$ $SmthGrpd_\infty$ and $X \twoheadrightarrow \mathcal{X}$ 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 $X_\bullet$, $Y_\bullet$ is a morphism between their associated $\infty$-stacks $\left\vert X_\bullet \right\vert \to \left\vert Y_\bullet\right\vert$, 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.