In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a monoid (or monoid object, or internal monoid, or algebra) in a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
is an
object ''M'' together with two
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s
* ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'',
* ''η'': ''I'' → ''M'' called ''unit'',
such that the pentagon
diagram
A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...
:
and the unitor diagram
:
commute. In the above notation, 1 is the
identity morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...
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
braiding ''γ''. A monoid ''M'' in C is commutative when .
Examples
* A monoid object in
Set, the
category of sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
(with the monoidal structure induced by the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
), is a
monoid in the usual sense.
* A monoid object in
Top, the
category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
(with the monoidal structure induced by the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
), 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 , the
category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
, is a
ring.
* For a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''R'', a monoid object in
** , 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
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, and a comonoid object is a ''K''-
coalgebra.
* For any category ''C'', the category of its
endofunctors has a monoidal structure induced by the composition and the identity
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
''I''
''C''. A monoid object in 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 . Dually in a category with an initial object and
finite coproducts every object becomes a monoid object via .
Categories of monoids
Given two monoids and in a monoidal category C, a morphism 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.
[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