Monoid (category Theory)

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category 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, is the identity morphism of , 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 .

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 an ''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 ''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 finite products, every object becomes a comonoid object via the diagonal morphism $\Delta_X: X\to X\times X$. Dually in a category with finite coproducts every object becomes a monoid object via $id_X\sqcup id_X: X\sqcup X \to X$.

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'' o ''μ'' = ''μ o (''f'' ⊗ ''f''),
* ''f'' o ''η'' = ''η.
In other words, the following diagrams

,

commute.

The category of monoids in C and their monoid morphisms is written Mon_C.Section VII.3 in

See also

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

References

*{{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

Monoidal categories
Objects (category theory)
Categories in category theory