# nLab singular-smooth ∞-groupoid

Contents

### Context

#### Representation theory

representation theory

geometric 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 $G$-fixed loci for any finite group $G$.

## Definition

Write

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

$SingSmoothGrpd_\infty \;\coloneqq\; Sh_\infty \big( CartesianSpaces \times Singularities \big) \,.$

## Properties

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

Where the cohesive $\infty$-topos $SmthGrpd_\infty$ is the natural home of smooth manifolds and diffeological spaces, reflecting their differential cohomology, so $SingSmoothGrpd_\infty$ 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.