(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
For a sheaf topos, a group object and any object, and for an action of on , the quotient stack is the quotient of this action but formed not in but under the inclusion
into the (2,1)-topos over the given site of definition: it is the quotient after regarding the action as an infinity-action in .
This is the geometric version of the notion of action groupoid. A wider notion of the quotient stack may be defined using more general internal groupoids. Indeed, the small fibration obtained by the externalization of an internal groupoid in a site with pullbacks will be a fibered category which is a candidate for a quotient stack in this context. For many interesting sites, sometimes under additional conditions on the internal groupoid, the resulting small fibration is indeed a stack.
If the stabilizer subgroups of the action are finite groups, then the quotient stack is an orbifold/Deligne-Mumford stack –the “quotient orbifold”.
Let be a Lie group action on a manifold (left action).
We define the quotient stack as
Morphisms of objects are -equivariant isomorphisms. This definition is taken from Heinloth’s Some notes on Differentiable stacks.
Given a Lie group action of on , if we want to associate a stack, we start with simpler cases which allows us to guess how to define in general.
Suppose is trivial and acts trivially on then should only depend on . We know what stack to associate for a Lie group i.e., . Thus, should just be .
Suppose is trivial and acts on , should only depend on . We know what stack to associate for a manifold i.e., . Thus, should just be .
Suppose is non trivial and is non trivial and that the action of on is free (and proper) so that is a manifold. We know what stack to associate for a manifold i.e., . Thus, should just be .
For general case of acting on , we get a Lie groupoid, called the Translation groupoid (or action groupoid) usually denoted by .
For action groupoid , let be the corresponding stack of principal bundles. It turns out that is same defined above. More details to be found in this page.
If action of the Lie group on the manifold is free and proper, what we get is a manifold . Stack associated to this manifold is which we call to be the quotient stack, denote by .
If the action of the Lie group on the manifold is not necessarily free and proper, what we get is a Lie groupoid denoted (among other symbols) by . Stack associated to this Lie groupoid is which we call to be the quotient stack, denote by .
(references for what the following paragraphs are getting at are listed below)
Let be a Lie group and be a manifold with a -action.
Supposing that acts freely and properly on , the quotient stack will be the stack . This action yields a principal -bundle of manifolds , which gives a morphism of stacks . We refer to this stack morphism as a principal -bundle of stacks.
More precisely, a stack morphism is said to be representable if given a manifold and a stack morphism , the fiber product is a manifold. A representable morphism of stacks is said to be a principal bundle of stacks if the map is a principal -bundle of manifolds. The stack morphism is a principal -bundle of stacks, since the map is a principal -bundle of manifolds.
The property “ is a principal -bundle” is the main ingredient in the definition of the quotient stack . Irrespective of whether or not acts freely and properly on , we still want to define a quotient stack as a stack such that is a principal -bundle of stacks in a “minimal” way.
The quotient stack of the action of on is a stack equipped with a principal -bundle of stacks such that any other principal -bundle of stacks factors through .
If acts freely and properly, then an obvious choice for is the stack . By the universal property, must be precisely the stack appearing in the definition of quotient stacks, i.e.
Morphisms of quotient stacks are isomorphisms of principal -bundles that commute with -equivariant morphisms. Fixing notation, we write for and refer to this as the quotient stack.
For the terminal object, one writes . This is the moduli stack for -principal bundles. It is also the trivial -gerbe.
There is a canonical projection . This is the universal rho-associated bundle.
The definition of quotient stacks as stackifications of (presheaves of) action groupoids is considered for instance in:
Gérard Laumon, Laurent Moret-Bailly, Prop. 4.3.1 in: Champs algébriques, Ergebn. der Mathematik und ihrer Grenzgebiete 39, Springer (2000) [doi:10.1007/978-3-540-24899-6]
Stacks Project, Definition 77.20.1 in: Quotient Stacks [tag:044Q]
The construction of quotient stacks as prestacks of -principal bundles equipped with -equivariant maps to is considered for instance in:
Jochen Heinloth, Exp. 1.5 in: Notes on differentiable stacks (2004) [pdf, pdf]
Frank Neumann, p. 28 in: Algebraic Stacks and Moduli of Vector Bundles, impa (2011) [pdf, pdf]
Michael Groechenig, Def. 5.23 in: Algebraic Stacks, Lecture notes (2014) [web, pdf, pdf]
Daniel Halpern-Leistner, Section 7.1 of: Moduli theory, Lecture notes (2020) [pdf, pdf]
Discussion of sufficient conditions for this construction to really yield a stack (instead of just a prestack):
The characterization of quotient stacks as fibrations over delooping stacks with homotopy fiber (cf. also the discussion at -action):
Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Exp. 4.4 in: Principal -bundles – General theory, J. Hom. Rel. Struc. 10 4 (2015) 749-801 [arXiv:1207.0248, doi:10.1007/s40062-014-0083-6]
Hisham Sati, Urs Schreiber, Prop. 2.79 in: Proper Orbifold Cohomology [arXiv:2008.01101]
Hisham Sati, Urs Schreiber, Prop. 0.2.1 (ii) & Prop. 3.2.76 in: Equivariant principal -bundles [arXiv:2112.13654]
See also:
Last revised on June 1, 2023 at 19:18:07. See the history of this page for a list of all contributions to it.