Editing Groupoid (section)

=== Group action ===
{{main|action groupoid}}
If the [[group (mathematics)|group]] <math>G</math> acts on the set {{tmath|1= X }}, then we can form the '''[[action groupoid]]''' (or '''transformation groupoid''') representing this [[Group action (mathematics)|group action]] as follows:
* The objects are the elements of {{tmath|1= X }};
* For any two elements <math>x</math> and <math>y</math> in {{tmath|1= X }}, the [[morphism]]s from <math>x</math> to <math>y</math> correspond to the elements <math>g</math> of <math>G</math> such that {{tmath|1= gx = y }};
* [[Function composition|Composition]] of morphisms interprets the [[binary operation]] of {{tmath|1= G }}.

More explicitly, the ''action groupoid'' is a small category with <math>\mathrm{ob}(C)=X</math> and <math>\mathrm{hom}(C)=G\times X</math> and with source and target maps <math>s(g,x) = x</math> and {{tmath|1= t(g,x) = gx }}. It is often denoted <math>G \ltimes X</math> (or <math>X\rtimes G</math> for a right action). Multiplication (or composition) in the groupoid is then {{tmath|1= (h,y)(g,x) = (hg,x) }}, which is defined provided {{tmath|1= y=gx }}.

For <math>x</math> in {{tmath|1= X }}, the vertex group consists of those <math>(g,x)</math> with {{tmath|1= gx=x }}, which is just the [[isotropy subgroup]] at <math>x</math> for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the [[Orbit (group theory)|orbit]] of the group action, and the groupoid is transitive if and only if the group action is [[Transitive group action|transitive]].

Another way to describe <math>G</math>-sets is the [[functor category]] {{tmath|1= [\mathrm{Gr},\mathrm{Set}] }}, where <math>\mathrm{Gr}</math> is the groupoid (category) with one element and [[isomorphism|isomorphic]] to the group {{tmath|1= G }}. Indeed, every functor <math>F</math> of this category defines a set <math>X=F(\mathrm{Gr})</math> and for every <math>g</math> in <math>G</math> (i.e. for every morphism in {{tmath|1= \mathrm{Gr} }}) induces a [[bijection]] <math>F_g</math> : {{tmath|1= X\to X }}. The categorical structure of the functor <math>F</math> assures us that <math>F</math> defines a <math>G</math>-action on the set {{tmath|1= G }}. The (unique) [[representable functor]] <math>F : \mathrm{Gr} \to \mathrm{Set}</math> is the [[Cayley's theorem|Cayley representation]] of {{tmath|1= G }}. In fact, this functor is isomorphic to <math>\mathrm{Hom}(\mathrm{Gr},-)</math> and so sends <math>\mathrm{ob}(\mathrm{Gr})</math> to the set <math>\mathrm{Hom}(\mathrm{Gr},\mathrm{Gr})</math> which is by definition the "set" <math>G</math> and the morphism <math>g</math> of <math>\mathrm{Gr}</math> (i.e. the element <math>g</math> of {{tmath|1= G }}) to the permutation <math>F_g</math> of the set {{tmath|1= G }}. We deduce from the [[Yoneda embedding]] that the group <math>G</math> is isomorphic to the group {{tmath|1= \{F_g\mid g\in G\} }}, a [[subgroup]] of the group of [[permutation group|permutation]]s of {{tmath|1= G }}.

==== Finite set ====
Consider the group action of <math>\mathbb{Z}/2</math> on the finite set <math>X = \{-2, -1, 0, 1, 2\}</math> where 1 acts by taking each number to its negative, so <math>-2 \mapsto 2</math> and {{tmath|1= 1 \mapsto -1 }}. The quotient groupoid <math>[X/G]</math> is the set of equivalence classes from this group action {{tmath|1= \{[0],[1],[2]\} }}, and <math>[0]</math> has a group action of <math>\mathbb{Z}/2</math> on it.{{fact|date=May 2025}}

==== Quotient variety ====
Any finite group <math>
G
</math> that maps to <math>
GL(n)
</math> gives a group action on the [[affine space]] <math>
\mathbb{A}^n
</math> (since this is the group of automorphisms). Then, a quotient groupoid can be of the form {{tmath|1= [\mathbb{A}^n/G] }}, which has one point with stabilizer <math> G </math> at the origin. Examples like these form the basis for the theory of [[orbifold]]s. Another commonly studied family of orbifolds are [[weighted projective space]]s <math>\mathbb{P}(n_1,\ldots, n_k)</math> and subspaces of them, such as [[Calabi–Yau manifold|Calabi–Yau orbifold]]s.