Editing Group action (section)

== Morphisms and isomorphisms between ''G''-sets ==
If {{math|''X''}} and {{math|''Y''}} are two {{math|''G''}}-sets, a ''morphism'' from {{math|''X''}} to {{math|''Y''}} is a function {{math|''f'' : ''X'' → ''Y''}} such that {{math|1=''f''(''g''⋅''x'') = ''g''⋅''f''(''x'')}} for all {{math|''g''}} in {{math|''G''}} and all {{math|''x''}} in {{math|''X''}}. Morphisms of {{math|''G''}}-sets are also called ''[[equivariant map]]s'' or {{math|''G''}}-''maps''.

The composition of two morphisms is again a morphism. If a morphism {{math|''f''}} is bijective, then its inverse is also a morphism. In this case {{math|''f''}} is called an ''[[isomorphism]]'', and the two {{math|''G''}}-sets {{math|''X''}} and {{math|''Y''}} are called ''isomorphic''; for all practical purposes, isomorphic {{math|''G''}}-sets are indistinguishable.

Some example isomorphisms:
* Every regular {{math|''G''}} action is isomorphic to the action of {{math|''G''}} on {{math|''G''}} given by left multiplication.
* Every free {{math|''G''}} action is isomorphic to {{math|''G'' × ''S''}}, where {{math|''S''}} is some set and {{math|''G''}} acts on {{math|''G'' × ''S''}} by left multiplication on the first coordinate. ({{math|''S''}} can be taken to be the set of orbits {{math|''X'' / ''G''}}.)
* Every transitive {{math|''G''}} action is isomorphic to left multiplication by {{math|''G''}} on the set of left cosets of some subgroup {{math|''H''}} of {{math|''G''}}. ({{math|''H''}} can be taken to be the stabilizer group of any element of the original {{math|''G''}}-set.)

With this notion of morphism, the collection of all {{math|''G''}}-sets forms a [[category theory|category]]; this category is a [[Grothendieck topos]] (in fact, assuming a classical [[metalogic]], this [[topos]] will even be Boolean).