Monoid (category theory)

In category theory, a branch of mathematics, a '''monoid''' (or '''monoid object''', or '''internal monoid''', or '''algebra''') {{nowrap|<math>(M, \mu, \eta)</math>}} in a monoidal category {{nowrap|<math>(\mathcal C,\otimes,I)</math>}} is an object <math>M</math> together with two morphisms * <math>\mu\colon M\otimes M\to M</math> called ''multiplication'', * <math>\eta\colon I\to M</math> called ''unit'', such that the pentagon diagram : Image:Monoid multiplication.svg and the unitor diagram : Image:Monoid unit svg.svg commute. In the above notation, <math>1</math> is the identity morphism of <math>M</math>, <math>I</math> is the unit element and <math>\alpha,\lambda</math> and <math>\rho</math> are respectively the associator, the left unitor and the right unitor of the monoidal category <math>\mathcal C</math>.

Dually, a '''comonoid''' in a monoidal category <math>\mathcal C</math> is a monoid in the dual category <math>\mathcal C^{\mathrm{op}}</math>.

Suppose that the monoidal category <math>\mathcal C</math> has a braiding <math>\gamma</math>. A monoid <math>M</math> in <math>\mathcal C</math> is '''commutative''' when {{nowrap|<math>\mu\circ\gamma=\mu</math>}}.

== Examples == * A monoid object in '''Set''', the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. In this context: ** the unit object <math>I</math> of the monoidal category can be taken to be any singleton. ** the multiplication <math>\mu</math> corresponds to the monoid operation in the usual sense. ** the unit <math>\eta</math> corresponds to the function that maps the single member of <math>I</math> to the identity element in the monoid. * A monoid object in '''Top''', the category of topological spaces (with the monoidal structure induced by the product topology), is a topological monoid. * A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. * A monoid object in the category of complete join-semilattices '''Sup''' (with the monoidal structure induced by the Cartesian product) is a unital quantale. * A monoid object in {{nowrap|('''Ab''', ⊗'''Z''', '''Z''')}}, the category of abelian groups, is a ring. * For a commutative ring ''R'', a monoid object in ** {{nowrap|(''R''-'''Mod''', ⊗''R'', ''R'')}}, the category of modules over ''R'', is a ''R''-algebra. ** the category of graded modules is a graded ''R''-algebra. ** the category of chain complexes of ''R''-modules is a differential graded algebra. * A monoid object in ''K''-'''Vect''', the category of ''K''-vector spaces (again, with the tensor product), is a unital associative ''K''-algebra, and a comonoid object is a ''K''-coalgebra. * For any category ''C'', the category {{nowrap|[''C'', ''C'']}} of its endofunctors has a monoidal structure induced by the composition and the identity functor ''I''''C''. A monoid object in {{nowrap|[''C'', ''C'']}} is a monad on ''C''. * For any category with a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism {{nowrap|Δ''X'' : ''X'' → ''X'' × ''X''}}. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via {{nowrap|id''X'' &sqcup; id''X'' : ''X'' &sqcup; ''X'' → ''X''}}.

== Categories of monoids == Given two monoids {{nowrap|(''M'', ''μ'', ''η'')}} and {{nowrap|(''M''′, ''μ''′, ''η''′)}} in a monoidal category '''C''', a morphism {{nowrap|''f'' : ''M'' → ''M''′}} is a '''morphism of monoids''' when * ''f'' ∘ ''μ'' = ''μ''′ ∘ (''f'' ⊗ ''f''), * ''f'' ∘ ''η'' = ''η''′. In other words, the following diagrams

File:Category monoids mu.svg, File:Category monoids eta.svg

commute.

The category of monoids in '''C''' and their monoid morphisms is written '''Mon''''''C'''.<ref>Section VII.3 in {{cite book|last1=Mac Lane|first1=Saunders|author-link=Saunders Mac Lane|title=Categories for the working mathematician|date=1988|publisher=Springer-Verlag|location=New York|isbn=0-387-90035-7|edition=4th corr. print.}}</ref>

== See also == * Act-S, the category of monoids acting on sets

==References== {{Reflist}} *{{cite book |first1=Mati |last1=Kilp |first2=Ulrich |last2=Knauer |first3=Alexander V. |last3=Mikhalov |title=Monoids, Acts and Categories |date=2000 |publisher=Walter de Gruyter |isbn=3-11-015248-7}}

Category:Monoidal categories Category:Objects (category theory) Category:Categories in category theory