(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 .
Let be a Lie group and be a manifold with a action on it.
Suppose acts freely, properly on then, we have mentioned that the quotient stack has to be the stack . The proper, free action of on gives a principal bundle . This gives a map of stacks . We call the map of stacks to be a principal bundle.
A map of stacks is said to be representable morphism if given a manifold and a map of stacks , the fiber product is a manifold.
A map of stacks is said to be a principal bundle if it is a representable morphism and the map of manifolds is a principal bundle.
It is easy to see that the map of stacks is a principal bundle as the map of manifolds is a principal bundle.
We see the property “ is a principal bundle” as main ingredient to define the quotient stack . Irrespective of acting freely and properly on , we want to define quotient stack as a stack such that is a principal bundle in minimal terms.
More precisely, by quotient stack of the action of on , we mean a stack that comes with a map of stacks that is a principal bundle (in the sense defined above) any map of stacks that is a principal bundle factors through this map .
If acts freely and properly, then obvious choice for is the stack .
Using the universal property, it turns out that has to be the stack in the definition of quotient stack
Morphisms of objects are -equivariant isomorphisms. We fix the notation for and call it 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.
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Jack Morava, Theories of anything (arXiv:1202.0684)
Last revised on June 14, 2020 at 03:08:23. See the history of this page for a list of all contributions to it.