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In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms

μ: M ⊗ M → M called multiplication,
η: I → M called unit,

such that the pentagon diagram

and the unitor diagram

commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C.

Dually, a comonoid in a monoidal category C is a monoid in the dual category C^op.

Suppose that the monoidal category C has a symmetry γ. A monoid M in C is commutative when μ ∘ γ = μ.

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.
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 (Ab, ⊗_Z, Z), the category of abelian groups, is a ring.
For a commutative ring R, a monoid object in
- (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 [C, C] of its endofunctors has a monoidal structure induced by the composition and the identity functor I_C. A monoid object in [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 Δ_X : X → X × X. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via id_X ⊔ id_X : X ⊔ X → X.

Categories of monoids

Given two monoids (M, μ, η) and (M′, μ′, η′) in a monoidal category C, a morphism f : M → M′ is a morphism of monoids when

f ∘ μ = μ′ ∘ (f ⊗ f),
f ∘ η = η′.

In other words, the following diagrams

,

commute.

The category of monoids in C and their monoid morphisms is written Mon_C.^[1]

References

^ Section VII.3 in Mac Lane, Saunders (1988). Categories for the working mathematician (4th corr. print. ed.). New York: Springer-Verlag. ISBN 0-387-90035-7.

Kilp, Mati; Knauer, Ulrich; Mikhalov, Alexander V. (2000). Monoids, Acts and Categories. Walter de Gruyter. ISBN 3-11-015248-7.

[1] Section VII.3 in Mac Lane, Saunders (1988). Categories for the working mathematician (4th corr. print. ed.). New York: Springer-Verlag. ISBN 0-387-90035-7.

[1]

@@ Line 1: / Line 1: @@
+{{Short description|Object in category theory mathematics}}
-In [[category theory]], a '''monoid''' (or '''monoid object''') <math>(M,\mu,\eta)</math> in a [[monoidal category]] '''C''' is an object ''M'' together with two [[morphism|morphisms]]
+{{for|the algebraic structure|Monoid}}
-* <math>\mu : M\otimes M\to M</math> called ''multiplication'',
-* and <math>\eta : I\to M</math> called ''unit'',
-such that the diagrams
+In [[category theory]], a branch of [[mathematics]], a '''monoid''' (or '''monoid object''', or '''internal monoid''', or '''algebra''') {{nowrap|(''M'', ''μ'', ''η'')}} in a [[monoidal category]] {{nowrap|('''C''', ⊗, ''I'')}} is an [[Object (category theory)|object]] ''M'' together with two [[morphism]]s
-[[Image:Monoid_mult.png]] and [[Image:Monoid_unit.png]]
+* ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'',
+* ''η'': ''I'' → ''M'' called ''unit'',
+such that the pentagon [[Diagram (category theory)|diagram]]
+: [[Image:Monoid multiplication.svg]]
+and the unitor diagram
+: [[Image:Monoid unit svg.svg]]
+[[Commutative diagram|commute]]. In the above notation, 1 is the [[identity morphism]] of ''M'', ''I'' is the unit element and ''α'', ''λ'' and ''ρ'' are respectively the associativity, the left identity and the right identity of the monoidal category '''C'''.
+Dually, a '''comonoid''' in a monoidal category '''C''' is a monoid in the [[dual category]] '''C'''<sup>op</sup>.
-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'''.
-Dually, a '''comonoid''' in a monoidal category '''C''' is a monoid in the [[dual category]] <math>\mathbf{C}^{\mathrm{op}}</math>.
+Suppose that the monoidal category '''C''' has a [[Symmetric monoidal category|symmetry]] ''γ''. A monoid ''M'' in '''C''' is '''commutative''' when {{nowrap|1=''μ'' ∘ ''γ'' = ''μ''}}.
-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
-:<math>\mu\circ\gamma=\mu</math>.
 == Examples ==
-* A monoid object in '''[[category of sets|Set]]''' (with the monoidal structure induced by the cartesian product) is a [[monoid]] in the usual sense.
+* A monoid object in '''[[category of sets|Set]]''', the [[category of sets]] (with the monoidal structure induced by the [[Cartesian product]]), is a [[monoid]] in the usual sense.
-* A monoid object in '''[[category of topological spaces|Top]]''' (with the monoidal structure induced by the [[product topology]]) is a [[topological monoid]].
+* A monoid object in '''[[category of topological spaces|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 complete join-semilattices '''[[Complete_lattice#Morphisms_of_complete_lattices|Sup]]''' (with the monoidal structure induced by the cartesian product) is a unital [[quantale]].
+* 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 [[Semilattice#Complete_semilattices|complete join-semilattices]] '''[[Complete lattice#Morphisms of complete lattices|Sup]]''' (with the monoidal structure induced by the Cartesian product) is a unital [[quantale]].
-* A monoid object in [[category of abelian groups|'''Ab''']] is a [[ring (mathematics)|ring]].
-* For a commutative ring ''R'', a monoid object in the [[category of modules|'''''R''-Mod''']] is an [[R-algebra|''R''-algebra]].
+* A monoid object in {{nowrap|('''[[category of abelian groups|Ab]]''', ⊗<sub>'''Z'''</sub>, [[integer|'''Z''']])}}, the [[category of abelian groups]], is a [[Ring (mathematics)|ring]].
+* For a [[commutative ring]] ''R'', a monoid object in
-* A monoid object in [[category of vector spaces|'''''K''-Vect''']] is a ''K''-[[algebra over a field|algebra]], a comonoid object is a ''K''-[[coalgebra]].
+** {{nowrap|([[category of modules|''R''-'''Mod''']], ⊗<sub>''R''</sub>, ''R'')}}, the [[category of modules]] over ''R'', is a [[unital associative algebra|''R''-algebra]].
-* 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''.
+** the category of [[Graded_ring#Graded_module|graded modules]] is a [[graded ring#Graded algebra|graded ''R''-algebra]].
+** the [[Chain_complex#Category_of_chain_complexes|category of chain complexes]] of ''R''-modules is a [[differential graded algebra]].
+* A monoid object in ''K''-'''Vect''', the [[category of vector spaces|category of ''K''-vector spaces]] (again, with the tensor product), is a unital associative ''K''-[[Algebra over a field|algebra]], and a comonoid object is a ''K''-[[coalgebra]].
+* For any category ''C'', the category {{nowrap|[''C'', ''C'']}} of its [[endofunctor]]s has a monoidal structure induced by the composition and the identity [[functor]] ''I''<sub>''C''</sub>. A monoid object in {{nowrap|[''C'', ''C'']}} is a [[monad (category theory)|monad]] on ''C''.
+* For any category with a terminal object and [[Product (category theory)|finite products]], every object becomes a comonoid object via the diagonal morphism {{nowrap|Δ<sub>''X''</sub> : ''X'' → ''X'' × ''X''}}. Dually in a category with an initial object and [[Coproduct|finite coproducts]] every object becomes a monoid object via {{nowrap|id<sub>''X''</sub> &sqcup; id<sub>''X''</sub> : ''X'' &sqcup; ''X'' → ''X''}}.
 == Categories of monoids ==
-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
+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''),
-* <math>f\circ\mu = \mu'\circ(f\otimes f)</math>,
-* <math>f\circ\eta = \eta'</math>.
+* ''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 <math>\mathbf{Mon}_\mathbf{C}</math>.
+The category of monoids in '''C''' and their monoid morphisms is written '''Mon'''<sub>'''C'''</sub>.<ref>Section VII.3 in {{cite book|last1=Mac Lane|first1=Saunders|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 ==
-* [[monoid]] (non-categorical definition)
 * [[Act-S]], the category of monoids acting on sets
 ==References==
+{{Reflist}}
-* Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, ''Monoids, Acts and Categories'' (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7
+*{{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:Category-theoretic categories]]
+[[Category:Monoidal categories]]
+[[Category:Objects (category theory)]]
+[[Category:Categories in category theory]]