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I am wondering about the following:
Let $Singularities$ denote the global orbit category of finite groups, i.e. simply the full sub-$2$-category of all $\infty$-groupoids on those of the form $\ast \!\sslash\! G$ for $G$ a finite group.
Regarded as an $\infty$-site with trivial coverage, this is a cohesive $\infty$-site. Therefore, given any $\infty$-topos $\mathbf{H}_{\subset}$ we obtain a new $\infty$-topos
$\mathbf{H} \;\coloneqq\; PSh_\infty(Singularities, \mathbf{H}_{\subset})$which has the following properties:
for each finite group $G$ there is the usual $\ast \!\sslash\! G \in \mathbf{H}$, but in addition there is an object to be denoted $\prec^G \in \mathbf{H}$ – to be thought of as the the “generic $G$-orbi-singularity”
(namely that arising as the image of the corresponding object in $Singularities$ under the Yoneda-embedding and passing along the inverse terminal geometric morphism of $\mathbf{H}_{\subset}$ )
it carries an adjoint triple of modalities
$\lt \;\;\dashv\;\; \subset \;\;\dashv\;\; \prec$to be read
$singular \dashv smooth \dashv orbisingular$such that (at least when $\mathbf{H}_{\subset}$ is itself cohesive):
$\lt(\prec^G) \simeq \ast$
(“the purely singular aspect of an orbi-singularity is a plain quotient of a point, hence a point”)
$\subset(\prec^G) \simeq \ast \!\sslash\! G$
(“the purely smooth aspect of an orbi-singularity is a homotopy quotient of a point)
$\prec(\prec^G) \simeq \prec^G$
(“an orbi-singularity is purely orbi-singular”)
$\,$
I am wondering about the converse:
Suppose an $\infty$-topos $\mathbf{H}$ is such that these three conditions hold (the first one without its parenthetical remark).
Can we conclude that $\mathbf{H}$ is of the form $PSh_\infty(Singularities, \mathbf{H}_{\subset})$?
If not, which axioms could be added to make it work?
Sorry, nothing to add. Just to comment that discussion and material in this area is getting spread out, e.g., most discussion is at orbifold cohomology and most exposition at the corresponding page orbifold cohomology.
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