nLab action groupoid

Idea

Given an action $\rho$ of a group $G$ on a set $S$, the action groupoid $S \sslash G$ 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.

Definition

In category theory

Given an action $\rho \colon S \times G \to S$ of a group $G$ on the set $S$, the action groupoid $S \sslash G$ (or, more precisely, $S \sslash_\rho G$) 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 $(g,s) \colon s \to g s$.

• The composite of $(g,s) \colon s \to g s = s'$ and $(g',s')\colon s' \to g's'$ is $(g' g, s) \colon s \to g' g s$.

Equivalently, we may define the action groupoid $S \sslash G$ to be the groupoid

$$\array{ && S \times G \\ & {}^{s \coloneqq p_1}\swarrow && \searrow^{t = \rho} \\ S &&&& S }$$

with composition

$$(S \times G) \times_{t,s} (S \times G) \simeq S \times G \times G \to S \times G$$

given by the product in $G$.

We can denote the morphisms in $S\sslash G$ by

$$S\sslash G \coloneqq \{s\stackrel{g}{\to} \rho(s,g) | s\in S, g\in G\}.$$

In (∞,1)-category theory

Definition

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

$$\array{ \cdots & \underoverset{\to}{\to}{\to} & X\times_{G_0}G\times_p G & \rightrightarrows & X\times_{G_0}G & \to & X }$$

such that the degree-wise projections give a simplicial map

$$\array{ \cdots & \underoverset{\to}{\to}{\to} & X\times_{G_0}G\times_p G & \rightrightarrows & X\times_{G_0}G & \to & X \\ && \downarrow && \downarrow && \downarrow^a \\ \cdots & \underoverset{\to}{\to}{\to} & G\times_p G & \rightrightarrows & G & \xrightarrow{p} & G_0 }$$

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 $X$.

Interpretations

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

$$\rho \colon \mathbf{B}G \to Sets$$

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 $S$.

As a pseudo colimit

$S\sslash G$ is the 2-colimit of $\rho$, i.e., the category of elements of $\rho$.

$$S\sslash G \simeq colim_{\mathbf{B}G} \rho \,.$$

The universal cocone consists of cells of the form

$$\array{ S &&\stackrel{\rho(g)}{\to}&& S \\ & \searrow &\stackrel{\simeq}{\Leftarrow}& \swarrow \\ && S\sslash G } \,,$$

where the 2-morphism is uniquely specified and in components given by $s \mapsto (s \stackrel{g}{\to} \rho(s,g))$.

As associated universal bundle

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 $S\sslash G$ is the pullback

$$\array{ S\sslash G &\to& Sets_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Sets } \,.$$

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 $p \colon S \rtimes G \to G$ 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.

As a stack

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 $S\sslash G$.

Properties

Relation to representation theory

The action groupoids $X\sslash G$ of a group $G$ come equipped with a canonical map to $\mathbf{B}G \simeq \ast \sslash G$. 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.

Action $\infty$-groupoid

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

$$\rho \colon \mathbf{B}G \to (\infty,1)Cat$$

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 $V\sslash G$ as the (∞,1)-categorical colimit over the action:

$$V\sslash G \coloneqq \lim_\to \rho \,.$$

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

$$\array{ V\sslash G &\to& Z \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat }$$

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 $* \to \mathbf{B}G$

$$\array{ V&\to& V\sslash G &\to& Z \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat } \,.$$

The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $V$, which is $V$ itself, as indicated.

Some comments

• If the action of a Lie group $G$ on the manifold $X$ is free and proper, what you get is a manifold $X/G$.

• If the action of a Lie group $G$ on the manifold $X$ is not necesssarily free and proper, what you get is a Lie groupoid, denoted (among other symbols) by $X\sslash G$.

Related concepts

• global quotient orbifold, quotient stack, homotopy quotient

• the groupoid cardinality of action groupoids is given by the class formula

References

• 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