Generalized principal bundles and quotient stacks

We generalize the notions of principal bundles and quotient stacks to the categorical contexts of sites and bisites. These important notions have been developed and fruitfully applied in different geometrical contexts, such as those of topological spaces, schemes and manifolds. We provide a common generalization of these concepts in the context of Grothendieck sites, which are categories endowed with a Grothendieck topology. Moreover, we generalize this theory one dimension higher introducing twodimensional principal bundles and quotient 2-stacks over a 2-category endowed with a two-dimensional Grothendieck topology.

To develop a notion of principal bundle that makes sense in a generic site, we consider morphisms that are equivariant with respect to the action of a fixed group object. And we introduce a notion of locally trivial morphism that involves pullbacks along morphisms of a covering family in place of the restrictions considered in the topological setting. We then use the obtained generalized principal bundles to explicitly construct quotient prestacks over the site. Furthermore, we prove that, if the topology is subcanonical and the category is nice enough, they are stacks. The proof of the required gluing conditions strongly relies on the fact that the objects of a subcanonical site can be expressed as colimits of the covering sieves over them. We generalize such a theory of principal bundles and quotient stacks one dimension higher for 2-categories endowed with a bitopology. We introduce a notion of two-dimensional principal bundle by upgrading the group object to an internal 2- group in the 2-category. We also introduce a concept of 2-locally trivial morphism involving iso-comma objects along the morphisms of a covering bisieve. We then use principal 2-bundles to explicitly construct trihomomorphisms that are the analogues of quotient stacks one dimension higher. To perform this construction we need to restrict to the case of a (2,1)-category.

Since none of the available notions of higher-dimensional stack was suitable for the constructed trihomomorphisms, we introduce a notion of 2-stack that involves trihomomorphisms from a 2-category endowed with a bitopology into the tricategory of bicategories. Our definition of 2-stack generalizes a characterization of stack due to Street. We then give a useful characterization of 2-stack in terms of explicit conditions that can be checked more easily in practice. The obtained conditions generalize the usual definition of stack one dimension higher. Finally, we prove that, if the bitopology is subcanonical and the (2,1)-category is nice enough, the higher dimensional analogues of quotient stacks that we have produced are 2-stacks. In order to prove the required gluing conditions, we show that the objects of a subcanonical bisite can be expressed as sigma-bicolimits of the covering bisieves over them.