  We model systems as objects in a certain ambient Grothendieck site with
additional structure. We introduce generalized sheaves, called virtual
manifolds. These sheaves reconstruct an internal groupoid which models
instances of the holonomy groupoid of foliations, but in the broader framework
of integrable, almost regular systems. A generalization of the Morita invariant
homotopy type to the setting of internal groupoids, furnishes a generalized
version of the Lusternik-Schnirelmann invariant for groupoids. This invariant,
referred to herein as geometric complexity, computes the number of transitive
components of the generalized holonomy groupoid, up to Morita invariant
homotopy. We introduce a so-called internal absolute geometry, as a bicategory
localization at the Morita homotopies and internal point groupoids, i.e. at the
absolute point. We extend the geometric complexity to a well-defined bifunctor
on this bicategory localization.
