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monoid A monoid in a monoidal category is defined to be 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.
Classical monoids are of course just monoids in Set with the cartesian product.
A group object or internal group in a category with binary products and a terminal object is an object in and arrows
(the unit map)
(the inverse map) and
(the multiplication map), such that the following diagrams commute:
(expressing the fact multiplication is associative),
(telling us that the unit map picks out an element that is a left and right identity), and
(telling us that the inverse map really does take an inverse), where we have let denote the composite .
Even if doesn't have all binary products, as long as products with (and the terminal object ) exist, then one can still speak of a group object in .
A (unital, non-commutative) ring is (equivalently)
a monoid internal to the monoidal category (Ab,) where denotes the tensor product of rings.
a category enriched over Ab with one object.
a ringoid with one object.
Here is the category of abelian groups, made into a monoidal category using the tensor product of abelian groups. A commutative (unital) ring is an abelian monoid object in Ab.
commutative ring object?
A biring is defined to be a commutative ring object in the opposite of the category of commutative rings (also known as the category of affine schemes).
A biring is hence a ring equipped with ring homomorphisms called coaddition: cozero: co-additive inverse: comultiplication: and the multiplicative counit:
satisfying the dual axioms of a commutative ring.
Equivalently, a biring is a commutative ring equipped with a lift of the functor
to a functor
Birings form a monoidal category thanks to the fact that functors of this form are closed under composition. A monoid object in this monoidal category is called a plethory. A plethory is an example of a Tall–Wraith monoid.
The most important example of a biring is , the ring of symmetric polynomials. This is actually a plethysm.
A bialgebra (or bigebra?) is both an algebra?algebra and a coalgebra?coalgebra, where the operations of either one are homomorphisms for the other. A bialgebra is one of the ingredients in the concept of Hopf algebra?Hopf algebra.
More precisely a bialgebra is a monoid?monoid in?in the category of coalgebras?. Equivalently, it is a comonoid?comonoid in?in the category of algebras?algebras. Equivalently, it is a monoid in the category of comonoids in Vect?Vect — or equivalently, a comonoid in the category of monoids in Vect?Vect.
More generally, a bimonoid?bimonoid in a monoidal category is a monoid in the category of comonoids in — or equivalently, a comonoid in the category of monoids in . So, a bialgebra is a bimonoid in .
A -bialgebra with multiplication , comultiplication , unit and counit is a Hopf algebra if there exists a -linear map
called the antipode or coinverse such that (as a map ). If an antipode exists then it is unique, just the way that if inverses exist in a monoid they are unique. The unit is group like, hence , therefore . By linearity of this implies that .