In
category theory, a branch of
mathematics, 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 \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left a ...
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
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
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
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yield ...
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
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
(with the monoidal structure induced by the
Cartesian product), is a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
in the usual sense.
* A monoid object in
Top, the
category of topological spaces (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-s ...
), is a
topological monoid.
* A monoid object in the category of monoids (with the
direct product of monoids) is just a
commutative monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...
. 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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, and a comonoid object is a ''K''-
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
.
* For any category ''C'', the category
'C'',''C''of its
endofunctors has a monoidal structure induced by the composition and the identity
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
''I''
''C''. A monoid object in
'C'',''C''is a
monad
Monad may refer to:
Philosophy
* Monad (philosophy), a term meaning "unit"
**Monism, the concept of "one essence" in the metaphysical and theological theory
** Monad (Gnosticism), the most primal aspect of God in Gnosticism
* ''Great Monad'', a ...
on ''C''.
* For any category with
finite products, every object becomes a comonoid object via the diagonal morphism
. Dually in a category with
finite coproducts every object becomes a monoid object via
.
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