(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
In higher generalization (categorification) of how all equivalence relations in a Grothendieck topos have effective quotients, so in an -sheaf -topos all “-equivalence relations”, constituted by -groupoid objects have effective -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 -sheaf -toposes.
In an -topos , let be a simplicial object and write
for the -quotient coprojection, i.e. induced universal morphism from to the -colimit. Then:
(groupoid objects in an -topos are effective)
If is a groupoid object in that it satisfies the groupoidal Segal conditions, then the -quotient coprojection (1) is an effective epimorphism in that equivalent to its Cech nerve:
Morever, this correspondence extends to morphisms:
(equivalence between groupoids and effective epimoirphisms)
In any -topos the correspondence of Prop. extends to an equivalence of -categories
between the groupoid objects in and the full sub--category of the arrow category on the effective epimorphisms.
Here the -functors are the restriction of which sends a simplicial object to the universal morphism from to its -colimit and sends a morphism to its Cech nerve.
This follows since the correspondence in both directions is computed by taking a Kan extension to followed by a restriction, and this identifies both sides with the same full subcategory of .
(interpretation in terms of -stacks equipped with atlases)
In as far as every object in an -topos may be thought of an an -stack, an effective epimorphism is an atlas for this -stack and its Cech nerve is the -groupoid that presents the -stack.
For example, if or and is such that the Cech nerve takes values in 0-truncated concrete objects (using that these are cohesive -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 -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 , is a morphism between their associated -stacks , and, conversely, Prop. says that a morphism of -stacks equipped with compatible atlases becomes equivalently a morphism of presenting groupoids.
See p. 27 in Sati Schreiber 2020.
Last revised on November 2, 2021 at 10:09:09. See the history of this page for a list of all contributions to it.