nLab singular-smooth ∞-groupoid

Contents

Context

Cohesive \infty-Toposes

Representation theory

Contents

Idea

In generalization of how a smooth ∞-groupoid is an ∞-groupoid equipped with generalized smooth structure modeled on Cartesian spaces with smooth functions between them (hence: on smooth manifolds), a singular-smooth \infty-groupoid carries geometric structure which, in addition to being smooth almost everywhere, may have orbi-singularities, in that it is locally modeled on Cartesian space regarded possibly as GG-fixed loci for any finite group GG.

Definition

Write

Then singular-smooth \infty-groupoids are the objects in the hypercomplete \infty -sheaf \infty -topos over the product of these sites:

SingSmoothGrpd Sh (CartesianSpaces×Singularities). SingSmoothGrpd_\infty \;\coloneqq\; Sh_\infty \big( CartesianSpaces \times Singularities \big) \,.

Properties

In generalization of how smooth \infty -groupoids form a cohesive \infty -topos over Grpd Grpd_\infty , so singular-smooth \infty-groupoids form a singular-cohesive \infty -topos.

Where the cohesive \infty -topos SmthGrpd SmthGrpd_\infty is the natural home of smooth manifolds and diffeological spaces, reflecting their differential cohomology, so SingSmoothGrpd SingSmoothGrpd_\infty is the natural home of orbifolds reflecting their proper orbifold cohomology (SaSc 2020).

References

Last revised on October 26, 2021 at 06:56:13. See the history of this page for a list of all contributions to it.