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Given an action $\rho$ of a group $G$ on a set $S$, the action groupoid $S//G$ is a bit like the quotient set $S/G$ (the set of $G$-orbits). But, instead of taking elements of $S$ in the same $G$-orbit as being equal in $S/G$, 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 $G$ on $S$ 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 $S \rtimes G$. The advantage of this is that it accords with the generalisation to the action of a group $G$ on a groupoid $S$, which is relevant to orbit space considerations, since if $G$ acts on a space $X$ it also acts on the fundamental groupoid of $X$; this is fully developed in “Topology and Groupoids”, Chapter 11.
Given an action $\rho : S \times G \to S$ of a group $G$ on the set $S$, the action groupoid $S//G$ (or, more precisely, $S//_\rho G$) is the groupoid for which:
an object is an element of $S$
a morphism from $s \in S$ to $s' \in S$ is a group element $g \in G$ with $g s = s'$. So, a general morphism is a pair $(g,s) : s \to g s$.
The composite of $(g,s) : s \to g s = s'$ and $(g',s'): s' \to g's'$ is $(g' g, s) : s \to g' g s$.
Equivalently, we may define the action groupoid $S//G$ to be the groupoid
with composition
given by the product in $G$.
We can denote the morphisms in $S//G$ by
Let $C$ be an ($\infty$,1)-category, let $G\in Grpd(C)$ be a groupoid object in $C$, let $X\in C$ be an object. Then the simplicial object
such that the degree-wise projections give a simplicial map
is called an action of $G$ on $X$. The colimit $colim\; X\times_{G_0}^{\times_\bullet}$ is called action $\infty$-groupoid of $G$ on $X$.
On top of the above explicit definitions, there are several useful ways to think of action groupoids.
Recall that the action $\rho$ is equivalently thought of as a functor
from the group $G$ regarded as a one-object groupoid, denoted $\mathbf{B}G$.
This functor sends the single object of $\mathbf{B}G$ to the set $S$.
$S//G$ is the 2-colimit of $\rho$, i.e., the category of elements of $\rho$.
The universal cocone consists of cells of the form
where the 2-morphism is uniquely specified and in components given by $s \mapsto (s \stackrel{g}{\to} \rho(s,g))$.
Let $Set_*$ be the category of pointed sets and $Sets_* \to Sets$ be the canonical forgetful functor. We can think of this as the “universal $Set$-bundle”.
Then $S//G$ 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 $G$ on the set $S$ gives rise to a morphism $p: S \rtimes G \to G$ 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 $G$ to presentations of the covering morphism of $G$ derived from the action of $G$ 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 $S//G$.
The action groupoids $X//G$ of a group $G$ come equipped with a canonical map to $\mathbf{B}G \simeq \ast //G$. Regarded via this map as objects in the slice of groupoids over $\mathbf{B}G$, 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 $G$ be an ∞-group in that $\mathbf{B}G$ is an ∞-groupoid with a single object. An action of $G$ on an (∞,1)-category is an (∞,1)-functor
to (∞,1)Cat. This takes the single object of $\mathbf{B}G$ to some $(\infty,1)$-category $V$.
Again we want to define the action groupoid $V//G$ as the (∞,1)-categorical colimit over the action:
By the result described here this is, as before, equivalent to the pullback of the “universal $(\infty,1)Cat$-bundle” $Z \to (\infty,1)Cat$, namely to the coCartesian fibration
classified by $\rho$ under the (∞,1)-Grothendieck construction. As before, we can continue a fiber sequence to the left by adjoining the $(\infty,1)$-categorical pullback along the point inclusion $* \to \mathbf{B}G$
The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $C$, which is $V$ itself, as indicated.
P.J. Higgins, 1971, “Categories and Groupoids”, van Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195.
R. Brown, “Topology and groupoids”, Booksurge, 2006, available from amazon. details
John Armstrong’s article, Groupoids (and more group actions)
John Baez, TWF 249