nLab
action groupoid

Context

Homotopy theory

Representation theory

Contents

Idea

Given an action ρ\rho of a group GG on a set SS, the action groupoid S//GS//G is a bit like the quotient set S/GS/G (the set of GG-orbits). But, instead of taking elements of SS in the same GG-orbit as being equal in S/GS/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 GG on SS 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 SGS \rtimes G. The advantage of this is that it accords with the generalisation to the action of a group GG on a groupoid SS, which is relevant to orbit space considerations, since if GG acts on a space XX it also acts on the fundamental groupoid of XX (see “Topology and Groupoids”, Chapter 11).

Definition

In category theory

Given an action ρ:S×GS\rho : S \times G \to S of a group GG on the set SS, the action groupoid S//GS//G (or, more precisely, S// ρGS//_\rho G) is the groupoid for which:

  • an object is an element of SS

  • a morphism from sSs \in S to sSs' \in S is a group element gGg \in G with gs=sg s = s'. So, a general morphism is a pair (g,s):sgs(g,s) : s \to g s.

  • The composite of (g,s):sgs=s(g,s) : s \to g s = s' and (g,s):sgs(g',s'): s \to g's' is (gg,s):sggs(g' g, s) : s \to g' g s.

Equivalently, we may define the action groupoid S//GS//G to be the groupoid

S×G s:=p 1 t=ρ S S \array{ && S \times G \\ & {}^{s := p_1}\swarrow && \searrow^{t = \rho} \\ S &&&& S }

with composition

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

given by the product in GG.

We can denote the morphisms in S//GS//G by

S//G:={sgρ(s,g)sS,gG}.S//G:=\{s\stackrel{g}{\to} \rho(s,g) | s\in S, g\in G\}.

In (∞,1)-category theory

Definition

Let CC be an (\infty,1)-category, let GGrpd(C)G\in Grpd(C) be a groupoid object in CC, let XCX\in C be an object. Then the simplicial object

...X× G 0G× pGX× G 0GX ... \stackrel{\to}{\stackrel{\to}{\to}} X\times_{G_0}G\times_p G \stackrel{\to}{\to} X\times_{G_0}G \to X

such that the degree-wise projections give a simplicial map

...X× G 0G× pG X× G 0G X a G× pG G p G 0\array{ ... X\times_{G_0}G\times_p G & \stackrel{\to}{\to} & X\times_{G_0}G & \to & X \\ \downarrow&&\downarrow&&\downarrow^a \\ G\times_p G & \stackrel{\to}{\to} & G & \xrightarrow{p} & G_0 }

is called an action of GG on XX. The colimit colimX× G 0 × colim\; X\times_{G_0}^{\times_\bullet} is called action \infty-groupoid of GG on XX.

Interpretations

On top of the above explicit definitions, there are several useful ways to think of action groupoids.

Recall that the action ρ\rho is equivalently thought of as a functor

ρ:BGSets \rho : \mathbf{B}G \to Sets

from the group GG regarded as a one-object groupoid, denoted BG\mathbf{B}G.

This functor sends the single object of BG\mathbf{B}G to the set SS.

As a pseudo colimit

S//GS//G is the 2-colimit of ρ\rho, i.e., the category of elements of ρ\rho.

S//Gcolim BGρ. S//G \simeq colim_{\mathbf{B}G} \rho \,.

The universal cocone consists of cells of the form

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

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

As associated universal bundle

Let Set *Set_* be the category of pointed sets and Sets *SetsSets_* \to Sets be the canonical forgetful functor. We can think of this as the “universal SetSet-bundle”.

Then S//GS//G is the pullback

S//G Sets * BG ρ Sets. \array{ S//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 GG on the set SS gives rise to a morphism p:SGGp: 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 GG to presentations of the covering morphism of GG derived from the action of GG 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//GS//G.

Action \infty-groupoid

All of this goes through almost verbatim for actions in the context of (∞,1)-category theory.

Let GG be an ∞-group in that BG\mathbf{B}G is an ∞-groupoid with a single object. An action of GG on an (∞,1)-category is an (∞,1)-functor

ρ:BG(,1)Cat \rho : \mathbf{B}G \to (\infty,1)Cat

to (∞,1)Cat. This takes the single object of BG\mathbf{B}G to some (,1)(\infty,1)-category VV.

Again we want to define the action groupoid V//GV//G as the (∞,1)-categorical colimit over the action:

C//G:=lim ρ. C//G := \lim_\to \rho \,.

By the result described here this is, as before, equivalent to the pullback of the “universal (,1)Cat(\infty,1)Cat-bundle” Z(,1)CatZ \to (\infty,1)Cat, namely to the coCartesian fibration

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

classified by ρ\rho under the (∞,1)-Grothendieck construction. As before, we can continue a fiber sequence to the left by adjoining the (,1)(\infty,1)-categorical pullback along the point inclusion *BG* \to \mathbf{B}G

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

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

References

  • 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.

  • John Armstrong’s article, Groupoids (and more group actions)

  • John Baez, TWF 249

Revised on April 3, 2014 10:42:09 by Anonymous Coward (134.76.62.130)