In mathematics, an action groupoid (or transformation groupoid) is a groupoid that encodes a group action.
Definition Given any right group action
X × G → X , {\displaystyle X\times G\to X,}
its action groupoid is the small category defined as follows:
the objects are elements of X {\displaystyle X} , the morphisms from x {\displaystyle x} to y {\displaystyle y} are the elements of X × G {\displaystyle X\times G} ; the composition between x → g y {\displaystyle x{\overset {g}{\to }}y} and y → h z {\displaystyle y{\overset {h}{\to }}z} is x → h g z {\displaystyle x{\overset {hg}{\to }}z} . Since a groupoid is often depicted using two arrows, the action groupoid can be written as
X × G ⇉ t s X {\displaystyle X\times G\,{\overset {s}{\underset {t}{\rightrightarrows }}}\,X}
where s , t {\displaystyle s,t} denote the source and the target of a morphism in G {\displaystyle {\mathcal {G}}} ; thus, s ( x , g ) = x {\displaystyle s(x,g)=x} is the projection and t ( x , g ) = x g {\displaystyle t(x,g)=xg} is the given group action. Moreover
the unit of x ∈ M {\displaystyle x\in M} is ( x , e ) {\displaystyle (x,e)} ; the inverse of ( x , g ) {\displaystyle (x,g)} is ( x g , g − 1 ) {\displaystyle (xg,g^{-1})} . The analogous definition can be given for left group actions.
Properties Several concepts related to a group action X × G → X {\displaystyle X\times G\to X} can be presented via its action groupoid G := X × G ⇉ X {\displaystyle {\mathcal {G}}:=X\times G\rightrightarrows X} :
the isotropy group G x := { g ∈ G | x g = x } ⊆ G {\displaystyle G_{x}:=\{g\in G\ |\ xg=x\}\subseteq G} at x ∈ X {\displaystyle x\in X} coincides with the isotropy group G x := G ( x , x ) {\displaystyle {\mathcal {G}}_{x}:={\mathcal {G}}(x,x)} of G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} at x ∈ X {\displaystyle x\in X} ; the orbit O x = { y ∈ M | ∃ g ∈ G with y = x g } ⊆ M {\displaystyle {\mathcal {O}}_{x}=\{y\in M\ |\ \exists g\in G{\text{ with }}y=xg\}\subseteq M} of x ∈ X {\displaystyle x\in X} coincides with the orbit s ( t − 1 ( x ) ) {\displaystyle s(t^{-1}(x))} of G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} at x ∈ X {\displaystyle x\in X} ; the orbit space M / G {\displaystyle M/G} of the group action coincides with the orbit space of G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} . As a consequence, a group action is transitive if and only if its action groupoid is transitive.
Topological setting If G {\displaystyle G} is a topological group and the G {\displaystyle G} -action is a continuous group action, then its action groupoid G := X × G ⇉ X {\displaystyle {\mathcal {G}}:=X\times G\rightrightarrows X} is a topological groupoid. In such case
the group action is proper if and only if G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} is proper;
G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} is source k {\displaystyle k} -connected if and only if G {\displaystyle G} is k {\displaystyle k} -connected;
Smooth setting If G {\displaystyle G} is a Lie group and the G {\displaystyle G} -action is a Lie group action, then its action groupoid G := X × G ⇉ X {\displaystyle {\mathcal {G}}:=X\times G\rightrightarrows X} is a Lie groupoid. In such case
G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} is étale if and only if G {\displaystyle G} is discrete;
G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} is effective if the G {\displaystyle G} -action is free and G {\displaystyle G} is discrete; if the group action is transitive, then G ⇉ X {\displaystyle {\mathcal {G}}\rightrightarrows X} is isomorphic to the gauge groupoid associated to the principal G x {\displaystyle G_{x}} -bundle G → O x = X {\displaystyle G\to {\mathcal {O}}_{x}=X} (for any point x ∈ X {\displaystyle x\in X} ). The Lie algebroid of the action groupoid G := X × G ⇉ X {\displaystyle {\mathcal {G}}:=X\times G\rightrightarrows X} is the action algebroid associated to the infinitesimal action of the Lie algebra g = L i e ( G ) {\displaystyle {\mathfrak {g}}=\mathrm {Lie} (G)} on X {\displaystyle X} .
In an ∞-category Let C {\displaystyle C} be an ∞-category and G {\displaystyle G} a groupoid object in it. Then a group action or an action groupoid on an object X {\displaystyle X} in C {\displaystyle C} is the simplicial diagram
⋯ ⇉ ⇉ X × G × G ⇉ → X × G ⇉ X {\displaystyle \cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,X\times G\times G\,{\underset {\rightarrow }{\rightrightarrows }}\,X\times G\,\rightrightarrows \,X}
that satisfies the axioms similar to an action groupoid in the usual case.
References
Works cited
Further reading https://ncatlab.org/nlab/show/action+groupoid https://mathoverflow.net/questions/130950/groupoids-vs-action-groupoids https://www.math.sci.hokudai.ac.jp/~wakate/mcyr/2023/pdf/uchimura_tomoki.pdf in Japanese