Group with operators: Difference between revisions

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In [[category theory]], a '''group with operators''' can be defined{{sfn|Mac Lane|1998|p=41}} as an [[object (category theory)|object]] of a [[functor category]] '''Grp'''^''M'' where ''M'' is a [[monoid]] (i.e. a [[category (mathematics)|category]] with one object) and '''Grp''' denotes the [[category of groups]]. This definition is equivalent to the previous one, provided <math>\Omega</math> is a monoid (if not, we may expand it to include the [[identity function|identity]] and all [[function composition|compositions]]).

A [[morphism]] in this category is a [[natural transformation]] between two [[functor]]s (i.e., two groups with operators sharing same operator ___domain ''M''&{{hairsp;}}). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the [[natural transformation#Definition|component]] of the natural transformation).

A group with operators is also a mapping

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*{{cite book | last=Mac Lane | first=Saunders | title=Categories for the Working Mathematician | publisher=Springer-Verlag | year=1998 | isbn=0-387-98403-8}}

[[Category:Group actions (mathematics)]]

[[Category:Universal algebra]]