nLab singular-smooth ∞-groupoid

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

• $CartesionSpaces \xhookrightarrow{\;} SmthMfd$ for the category of Cartesian spaces with smooth functions between them, regarded as a site via the coverage of differentiably good open covers,

• $Singularities \xhookrightarrow{\;} Grpd_\infty$ for the (2,1)-category of groupoids which are deloopings of finite groups, regarded as an $\infty$-site via the trivial topology.

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).

Related concepts

• orbifold, orbifold cohomology

• singular-smooth $\infty$-topos

• cohesion of global- over G-equivariant homotopy theory

References

• Hisham Sati, Urs Schreiber, Ex. 3.56 in: Proper Orbifold Cohomology (arXiv:2008.01101)