Equivalence of complete distributive inverse semigroups and étale localic groupoids, and a characterization of topological orbifolds

Abstract

As the main result of this dissertation, we established an equivalence between the category of complete distributive inverse semigroups and that of etale localic groupoids by introducing a new class of morphisms for the former. This equivalence puts many similar classical results into a firm ground while allowing us to describe topological and Lie groupoids, whose structure maps are local homeomorphisms and local diffeomorphisms, respectively, in terms of complete distributive inverse semigroups. Secondly, we computed the weak equivalences analogues to fully faithful essentially surjective etale functors for complete distributive inverse semigroups and use them to advance the previous equivalence to that of two relative categories. The localization of etale localic groupoids with respect to Morita equivalences are known as etale stacks, and this is also equivalent to etendues. The corresponding bicategory of fractions of complete distributive inverse semigroups allowed us to study etale stacks and geometric structures on them via inverse semigroups. In particular, we received a new characterization for topological orbifolds. Finally, we studied complete distributive inverse semigroups represented by Lie groupoids as those who have a smooth structure (i.e., a structure sheaf which is locally isomorphic to smooth functions on R^n) on the set of idempotents. After the bicategorical localization, these are precisely the inverse semigroup version of differentiable stacks.

Embargo status: Restricted until September 2022. To request an access exception, click on the PDF link to the left.

Embargo status: Restricted until September 2022. To request an access exception, click on the PDF link to the left.