Backround
Definition
Presentation over a site
Models
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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 $G$.
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:
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).
Last revised on October 26, 2021 at 10:56:13. See the history of this page for a list of all contributions to it.