Editing Equivariant map (section)

==Formalization==
Equivariance can be formalized using the concept of a [[Group action (mathematics)|{{mvar|G}}-set]] for a [[group (mathematics)|group]] {{mvar|G}}. This is a mathematical object consisting of a [[set (mathematics)|mathematical set]] {{mvar|S}} and a [[Group action (mathematics)|group action]] (on the left) of {{mvar|G}} on {{mvar|S}}.
If {{mvar|X}} and {{mvar|Y}} are both {{mvar|G}}-sets for the same group {{mvar|G}}, then a function {{math|''f'' : ''X'' → ''Y''}} is said to be equivariant if
:{{math|1=''f''(''g''&middot;''x'') = ''g''&middot;''f''(''x'')}}
for all {{math|''g'' ∈ ''G''}} and all {{math|''x'' in ''X''}}.<ref>{{citation|title=Nominal Sets: Names and Symmetry in Computer Science|volume=57|series=Cambridge Tracts in Theoretical Computer Science|first=Andrew M.|last=Pitts|publisher=Cambridge University Press|year=2013|isbn=9781107244689|url=https://books.google.com/books?id=VVehscCSPh8C&pg=PA14|at=Definition 1.2, p.&nbsp;14}}.</ref>

If one or both of the actions are right actions the equivariance condition may be suitably modified:
:{{math|1=''f''(''x''&middot;''g'') = ''f''(''x'')&middot;''g''}}; (right-right)
:{{math|1=''f''(''x''&middot;''g'') = ''g''<sup>&minus;1</sup>&middot;''f''(''x'')}}; (right-left)
:{{math|1=''f''(''g''&middot;''x'') = ''f''(''x'')&middot;''g''<sup>&minus;1</sup>}}; (left-right)

Equivariant maps are [[homomorphism]]s in the [[Category (mathematics)|category]] of ''G''-sets (for a fixed ''G'').<ref name="grm">{{citation|title=Groups, Rings, Modules|series=Dover Books on Mathematics|first1=Maurice|last1=Auslander|first2=David|last2=Buchsbaum|publisher=Dover Publications|year=2014|isbn=9780486490823|url=https://books.google.com/books?id=VW2TAwAAQBAJ&pg=PA86|pages=86–87}}.</ref> Hence they are also known as '''''G''-morphisms''',<ref name="grm"/> '''''G''-maps''',<ref>{{citation
 | last = Segal | first = G. B.
 | contribution = Equivariant stable homotopy theory
 | mr = 0423340
 | pages = 59–63
 | publisher = Gauthier-Villars, Paris
 | title = Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2
 | year = 1971}}.</ref> or '''''G''-homomorphisms'''.<ref>{{citation
 | last1 = Adhikari | first1 = Mahima Ranjan
 | last2 = Adhikari | first2 = Avishek
 | doi = 10.1007/978-81-322-1599-8
 | isbn = 978-81-322-1598-1
 | location = New Delhi
 | mr = 3155599
 | page = 142
 | publisher = Springer
 | title = Basic modern algebra with applications
 | url = https://books.google.com/books?id=lBO7BAAAQBAJ&pg=PA142
 | year = 2014}}.</ref> [[Isomorphism]]s of ''G''-sets are simply [[bijective]] equivariant maps.<ref name="grm"/>

The equivariance condition can also be understood as the following [[commutative diagram]].  Note that <math>g\cdot</math> denotes the map that takes an element <math>z</math> and returns <math>g\cdot z</math>.

[[Image:equivariant commutative diagram.png|center|175px]]