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Given an action of a group on a set , the action groupoid is a bit like the quotient set (the set of -orbits). But, instead of taking elements of in the same -orbit as being equal in , in the action groupoid they are just isomorphic. We may think of the action groupoid as a resolution of the usual quotient. When the action of on fails to be free, the action groupoid is generally better-behaved than the quotient set.
The action groupoid also goes by other names, including ‘weak quotient’. It is a special case of a ‘pseudo colimit’, as explained below. It is also called a “semidirect product” and then written . The advantage of this is that it accords with the generalisation to the action of a group on a groupoid , which is relevant to orbit space considerations, since if acts on a space it also acts on the fundamental groupoid of ; this is fully developed in “Topology and Groupoids”, Chapter 11.
Given an action of a group on the set , the action groupoid (or, more precisely, ) is the groupoid for which:
an object is an element of
a morphism from to is a group element with . So, a general morphism is a pair .
The composite of and is .
Equivalently, we may define the action groupoid to be the groupoid
with composition
given by the product in .
We can denote the morphisms in by
Let be an (,1)-category, let be a groupoid object in , let be an object. Then the simplicial object
such that the degree-wise projections give a simplicial map
is called an action of on . The colimit is called action -groupoid of on .
On top of the above explicit definitions, there are several useful ways to think of action groupoids.
Recall that the action is equivalently thought of as a functor
from the group regarded as a one-object groupoid, denoted .
This functor sends the single object of to the set .
is the 2-colimit of , i.e., the category of elements of .
The universal cocone consists of cells of the form
where the 2-morphism is uniquely specified and in components given by .
Let be the category of pointed sets and be the canonical forgetful functor. We can think of this as the “universal -bundle”.
Then is the pullback
One place where we discussed this is the comment It was David Roberts who apparently first noticed….
Notice also that an action of on the set gives rise to a morphism which has the property of unique path lifting, or in other words is a discrete opfibration. It is also called a covering morphism of groupoids, and models nicely covering maps of spaces.
Higgins used this idea to lift presentations of a group to presentations of the covering morphism of derived from the action of on cosets, and so to apply graph theory to obtain old and new subgroup theorems in group theory.
In the case where the action is internal to sets with structure, such as internal to Diff one wants to realize the action groupoid as a Lie groupoid. That Lie groupoid in turn may be taken to present a differentiable stack which then usually goes by the same name .
The action groupoids of a group come equipped with a canonical map to . Regarded via this map as objects in the slice of groupoids over , action groupoids are in fact equivalent to the actions that they arise from.
For more on this see at infinity-action and also at geometry of physics – representations and associated bundles.
All of this goes through almost verbatim for actions in the context of (∞,1)-category theory.
Let be an ∞-group in that is an ∞-groupoid with a single object. An action of on an (∞,1)-category is an (∞,1)-functor
to (∞,1)Cat. This takes the single object of to some -category .
Again we want to define the action groupoid as the (∞,1)-categorical colimit over the action:
By the result described here this is, as before, equivalent to the pullback of the “universal -bundle” , namely to the coCartesian fibration
classified by under the (∞,1)-Grothendieck construction. As before, we can continue a fiber sequence to the left by adjoining the -categorical pullback along the point inclusion
The resulting total -pullback rectangle is the fiber of over the -category , which is itself, as indicated.
If the action of a Lie group on the manifold is free and proper, what you get is a manifold .
If the action of a Lie group on the manifold is not necesssarily free and proper, what you get is a Lie groupoid, denoted (among other symbols) by .
the groupoid cardinality of action groupoids is given by the class formula
P.J. Higgins, 1971, “Categories and Groupoids”, van Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195.
Ronnie Brown, Topology and groupoids, Booksurge (2006) [webpage, pdf]
John Armstrong’s article, Groupoids (and more group actions)
John Baez, TWF 249
Last revised on July 14, 2024 at 18:30:21. See the history of this page for a list of all contributions to it.