In mathematics, an action groupoid or a transformation groupoid is a

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

that expresses a

group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...

. Namely, given a (right) group action :

X \times G \to X,

we get the groupoid

\mathcal

(= a category whose morphisms are all invertible) where *objects are elements of

X

, *morphisms from

x

y

are the actions of elements

g

G

such that

y = xg

, *compositions for

x \overset\to y

and

y \overset\to z

x \overset\to z

. A groupoid is often depicted using two arrows. Here the above can be written as: :

X \times G \,\overset\underset\rightrightarrows\, X

where

s, t

denote the source and the target of a morphism in

\mathcal

; thus,

s(x, g) = x

is the projection and

t(x, g) = xg

is the given group action (here the set of morphisms in

\mathcal

is identified with

X \times G

In an ∞-category

Let

C

be an ∞-category and

G

groupoid object In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Defini ...

in it. Then a group action or an action groupoid on an object ''X'' in ''C'' is the

simplicial diagram In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all = \ and the order-preserving functions). Formally, a simplicial diagram in a category or an ∞-cat ...

\cdots \, \underset\rightrightarrows \, X \times G \times G \, \underset\rightrightarrows \, X \times G \, \rightrightarrows\, X

that satisfies the axioms similar to an action groupoid in the usual case.

In an ∞-category

References

Works cited

Further reading