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| In [[category theory]], a '''monoid''' (or '''monoid object''') <math>(M,\mu,\eta)</math> in a [[monoidal category]] <math>(\mathbf{C}, \otimes, I)</math> is an object ''M'' together with two [[morphism]]s
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| * <math>\mu : M\otimes M\to M</math> called ''multiplication'',
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| * and <math>\eta : I\to M</math> called ''unit'',
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| such that the pentagon diagram
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| :[[Image:Monoid mult.png]]
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| and the unitor diagram | |
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| :[[Image:Monoid unit.png]]
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| commute. In the above notations, ''I'' is the unit element and <math>\alpha</math>, <math>\lambda</math> and <math>\rho</math> are respectively the associativity, the left identity and the right identity of the monoidal category '''C'''.
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| Dually, a '''comonoid''' in a monoidal category '''C''' is a monoid in the [[dual category]] <math>\mathbf{C}^{\mathrm{op}}</math>.
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| Suppose that the monoidal category '''C''' has a [[symmetric monoidal category|symmetry]] <math>\gamma</math>. A monoid <math>M</math> in '''C''' is '''symmetric''' when
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| :<math>\mu\circ\gamma=\mu</math>.
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| == Examples ==
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| * A monoid object in '''[[category of sets|Set]]''' (with the monoidal structure induced by the cartesian product) is a [[monoid]] in the usual sense.
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| * A monoid object in '''[[category of topological spaces|Top]]''' (with the monoidal structure induced by the [[product topology]]) is a [[topological monoid]].
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| * 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 theorem]].
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| * A monoid object in the category of complete join-semilattices '''[[Complete_lattice#Morphisms_of_complete_lattices|Sup]]''' (with the monoidal structure induced by the cartesian product) is a unital [[quantale]].
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| * A monoid object in ('''[[category of abelian groups|Ab]]''', ⊗<sub>'''Z'''</sub>, '''Z''') is a [[ring (mathematics)|ring]].
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| * For a commutative ring ''R'', a monoid object in ('''[[category of modules|''R''-Mod]]''', ⊗<sub>''R''</sub>, ''R'') is an [[R-algebra|''R''-algebra]].
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| * A monoid object in '''[[category of vector spaces|''K''-Vect]]''' (again, with the tensor product) is a ''K''-[[algebra over a field|algebra]], a comonoid object is a ''K''-[[coalgebra]].
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| * For any category ''C'', the category ''[C,C]'' of its [[endofunctor]]s has a monoidal structure induced by the composition. A monoid object in ''[C,C]'' is a [[monad (category theory)|monad]] on ''C''.
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| == Categories of monoids ==
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| Given two monoids <math>(M,\mu,\eta)</math> and <math>(M',\mu',\eta')</math> in a monoidal category '''C''', a morphism <math>f:M\to M'</math> is a '''morphism of monoids''' when
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| * <math>f\circ\mu = \mu'\circ(f\otimes f)</math>,
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| * <math>f\circ\eta = \eta'</math>.
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| The category of monoids in '''C''' and their monoid morphisms is written <math>\mathbf{Mon}_\mathbf{C}</math>.
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| == See also ==
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| * [[monoid]] (non-categorical definition)
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| * [[Act-S]], the category of monoids acting on sets
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| ==References==
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| * Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, ''Monoids, Acts and Categories'' (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7
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| [[Category:Monoidal categories]]
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| [[Category:Objects (category theory)]]
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| [[Category:Category-theoretic categories]]
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