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Generalized principal bundles and quotient stacks. (English) Zbl 1514.18012

This paper considers an internalized notion of principal bundle making sense in any site, provided that the underlying category has all pullbacks and a terminal object. The topological group involved in the standard notion becomes here a group object of the category and the notion of locally trivial morphism is internalized by considering pullbacks along the morphisms of a covering family for the Grothendieck topology of the site. This internalized notion coincides with the classical one when the site is \(\left(\mathcal{T}op,std\right)\), where \(\mathcal{T}op\) is the category of compactly generated Hausdorff topological spaces and \(std\) is the standard Grothendieck topology. The principal bundles in algebraic geometry are an instance of this notion.
The significane of principal bundles lies in the fact that they are the objects classified by classical spaces via homotopy pullbacks [J. W. Milnor and J. D. Stasheff, Characteristic classes. Princeton, NJ: Princeton University Press (1974; Zbl 0298.57008)] and by classifying stacks via \(2\)-pullbacks. Classifying stacks are particular cases of the very important notion of quotient stacks, which are commonly thought as stackification of presheaves of action groupoids [G. Laumon and L. Moret-Bailly, Champs algébriques. Berlin: Springer (2000; Zbl 0945.14005)], while an explicit construction using presheaves of groupoids of principal bundles rigged out in equivariant maps to a fixed space is present in the algebraic context [F. Neumann, Algebraic stacks and moduli of vector bundles. Paper from the 27th Brazilian Mathematics Colloquium – 27°Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 27–31, 2009. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (2009; Zbl 1177.14002)] and in the differential context [J. Heinloth, in: Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005. Lecture notes from the seminars “Number theory”, “Algebraic geometry” and “Twisted cohomology theories” held at the University of Göttingen, Göttingen, Germany, 2004. Göttingen: Universitätsverlag Göttingen. 1–32 (2005; Zbl 1098.14501)]. This paper gives the same construction in the internal context to define quotient prestacks, establishing that these objects are well defined pseudofunctors. The author finds sufficient abstract conditions guaranteeing that these generalized quotient prestacks are to abide by descent in the sense of stacks, which is the main result of this paper.
The synopsis of the paper goes as follows.
§ 2
recalls the notions of action and equivariant morphism in the context of group objects of a category, constructing an action of an internal group on a pullback with its actions on the sources of the two morphisms involved in the pullback given. The notion of locally trivial morphism in a site is introduced, and it is used to define principal bundles and morphisms between them. The section is concluded by observing that the notion truly generalizes the classical notion of principal bundles over a topological space.
§ 3
gives the definition of generalized quotient prestacks. A key result of the section is that generalized principal bundles are stable under pullbacks, being crucial to prove that the definition is a good one.
§ 4
recalls some useful results about the canonical topology, establishing the main result of the paper claiming that, under certain mild assumptions on the underlying category, the quoitent prestacks are stacks with respect to every subcanonical topology.

MSC:

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18F10 Grothendieck topologies and Grothendieck topoi
18C40 Structured objects in a category (group objects, etc.)
14A20 Generalizations (algebraic spaces, stacks)
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
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References:

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[25] Tim Van der Linden, Université catholique de Louvain: tim.vanderlinden@uclouvain.be
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