Contents
Context
Categorical algebra
Monoidal categories
monoidal categories
With symmetry
With duals for objects
With duals for morphisms
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Monoid theory
monoid theory in algebra:
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monoid, infinity-monoid
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monoid object, monoid object in an (infinity,1)-category
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Mon, CMon
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monoid homomorphism
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trivial monoid
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submonoid, quotient monoid?
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divisor, multiple?, quotient element?
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inverse element, unit, irreducible element
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ideal in a monoid
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principal ideal in a monoid
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commutative monoid
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cancellative monoid
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GCD monoid
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unique factorization monoid
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Bézout monoid
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principal ideal monoid
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group, abelian group
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absorption monoid
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free monoid, free commutative monoid
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graphic monoid
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monoid action
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module over a monoid
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localization of a monoid
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group completion
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endomorphism monoid
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super commutative monoid
Contents
Idea
Generalizing the classical notion of monoid, one can define a monoid (or monoid object) in any monoidal category . Classical monoids are of course just monoids in Set with the cartesian product.
By the microcosm principle, in order to define monoid objects in , itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if is a multicategory. Contrast this with a group object, which can only be defined in a cartesian monoidal category (or a cartesian multicategory).
Definition
Namely, a monoid in is an object equipped with a multiplication and a unit satisfying the associative law:
and the left and right unit laws:
Here is the associator in , while and are the left and right unitors.
Morphism of monoids
The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, between two monoid objects, satisfying the equations;
corresponding to the commutative diagrams;
As categories with one object
Just as the category of regular monoids is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in (with the obvious morphisms) is equivalent to the category of -enriched categories with one object.
Properties
Preservation by lax monoidal functors
Monoid structure is preserved by lax monoidal functors. Comonoid structure by oplax monoidal functors. See lax monoidal functor for more details.
Category of monoids
For special properties of categories of monoids, see category of monoids.
Examples
- A monoid object in Ab (with the usual tensor product of -modules as the tensor product) is a ring. A monoid object in the category of vector spaces over a field (with the usual tensor product of vector spaces) is an algebra over .
- A monoid in a category of modules is an associative unital algebra.
- A monoid object in Top (with cartesian product as the tensor product) is a topological monoid.
- A monoid object in Ho(Top) is an H-monoid?.
- A monoid object in the category of monoids (with cartesian product as the tensor product) is a commutative monoid. This is a version of the Eckmann-Hilton argument.
- A monoid object in the category of complete join-semilattices (with its tensor product that represents maps preserving joins in each variable separately) is a unital quantale.
- The category of pointed sets has a monoidal structure given by the smash product. A monoid object in this monoidal category is an absorption monoid.
- Given any monoidal category , a monoid in the monoidal category is called a comonoid in .
- In a cocartesian monoidal category, every object is a monoid object in a unique way.
- For any category , the endofunctor category has a monoidal structure induced by composition of endofunctors, and a monoid object in is a monad on .
These are examples of monoids internal to monoidal categories. More generally, given any bicategory and a chosen object , the hom-category has the structure of a monoidal category. So, the concept of monoid makes sense in any bicategory : we define a monoid in to be a monoid in for some object . This often called a monad in . The reason is that a monad in Cat is the same as monad on a category.
A monoid in a bicategory may also be described as the hom-object of a -enriched category with a single object.
References
Original reference (including the case of a commutative monoid in a symmetric monoidal category):
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Jean Bénabou, Algèbre élémentaire dans les catégories (1964), C. R. Acad. Sci. Paris 258 (1964) pp.771-774, gallica
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Saunders MacLane, Section III.6 in: Categories for the Working Mathematician, Springer (1971)
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Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 7 in: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)
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Florian Marty, Dections 1.2, 1.3 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009 (web)
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Martin Brandenburg, Section 4.1 of: Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)
See also MO/180673.
In cartesian monoidal categories:
and here formalized as mathematical structures in proof assistants:
in a context of plain Agda:
in a context of cubical Agda: