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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2020
    • (edited Jun 8th 2020)

    I am wondering about the following:

    Let SingularitiesSingularities denote the global orbit category of finite groups, i.e. simply the full sub-22-category of all \infty-groupoids on those of the form *G\ast \!\sslash\! G for GG a finite group.

    Regarded as an \infty-site with trivial coverage, this is a cohesive \infty-site. Therefore, given any \infty-topos H \mathbf{H}_{\subset} we obtain a new \infty-topos

    HPSh (Singularities,H ) \mathbf{H} \;\coloneqq\; PSh_\infty(Singularities, \mathbf{H}_{\subset})

    which has the following properties:

    1. for each finite group GG there is the usual *GH\ast \!\sslash\! G \in \mathbf{H}, but in addition there is an object to be denoted GH\prec^G \in \mathbf{H} – to be thought of as the the “generic GG-orbi-singularity”

      (namely that arising as the image of the corresponding object in SingularitiesSingularities under the Yoneda-embedding and passing along the inverse terminal geometric morphism of H \mathbf{H}_{\subset} )

    2. it carries an adjoint triple of modalities

      <\lt \;\;\dashv\;\; \subset \;\;\dashv\;\; \prec

      to be read

      singularsmoothorbisingularsingular \dashv smooth \dashv orbisingular
    3. such that (at least when H \mathbf{H}_{\subset} is itself cohesive):

      1. <( G)*\lt(\prec^G) \simeq \ast

        (“the purely singular aspect of an orbi-singularity is a plain quotient of a point, hence a point”)

      2. ( G)*G\subset(\prec^G) \simeq \ast \!\sslash\! G

        (“the purely smooth aspect of an orbi-singularity is a homotopy quotient of a point)

      3. ( G) G\prec(\prec^G) \simeq \prec^G

        (“an orbi-singularity is purely orbi-singular”)

    \,

    I am wondering about the converse:

    Suppose an \infty-topos H\mathbf{H} is such that these three conditions hold (the first one without its parenthetical remark).

    Can we conclude that H\mathbf{H} is of the form PSh (Singularities,H )PSh_\infty(Singularities, \mathbf{H}_{\subset})?

    If not, which axioms could be added to make it work?

    diff, v7, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 9th 2020

    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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