Internal absolute geometry I: desingularization
Karsten Bohlen
TL;DR
The paper develops an ambient, axiomatic framework for internal absolute geometry by embedding a structured Grothendieck site $( ext{A}, ext{T}, ext{E})$ into a diffeological setting via $oldsymbol{ extPhi}$, and showing how holonomy-like groupoids can be internal to the site. It defines $ ext{$ ext{T}$}$-local internal groupoids and spanoids, along with a Morita-theoretic calculus of bibundles and composition, enabling the reconstruction of integrated groupoid data from local atlas information. By constructing virtual manifolds as sheaves and equipping quotients with a path holonomy diffeology $ ext{$ ext{P}$}( ext{$oldsymbol{ extPsi}$}_{ ext{$ ext{U}$}})$, the work proves that the sheafification yields internal groupoids in the site, capturing desingularization data such as those arising from Nash blowups and Debord foliations. This framework unifies holonomy-type descriptions across singular foliations and moduli-type spaces, providing a local-to-global toolkit for desingularization and atlas reconstruction within a bicategorical, diffeological context.
Abstract
We introduce an axiomatization of Grothendieck sites with additional structure, and we describe sheaves that reconstruct groupoids which are internal to the site structure. This setting applies to various concrete situations, where a Nash blowup of a singular space results in an almost regular foliation. It also potentially applies to various types of moduli spaces. The sheaf can encode candidate holonomy groupoids that desingularize such spaces.