Spahn coalgebras, corings and birings in the theory of group shemes (Rev #1, changes)

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Monoid

monoid A monoid in a monoidal category CC is defined to be an object MM equipped with a multiplication μ:MMM\mu: M \otimes M \to M and a unit η:IM\eta: I \to M satisfying the associative law:

A pic

and the left and right unit laws:

A pic

Here α\alpha is the associator in CC, while λ\lambda and ρ\rho are the left and right unitors.

Classical monoids are of course just monoids in Set with the cartesian product.

Comonoid

Bimonoid

Group

group object

A group object or internal group in a category CC with binary products and a terminal object ** is an object GG in CC and arrows

1:*G 1:* \to G

(the unit map)

() 1:GG (-)^{-1}:G\to G

(the inverse map) and

m:G×GG m:G\times G \to G

(the multiplication map), such that the following diagrams commute:

G×G×G id×m G×G m×id m G×G m G \array{ G\times G\times G & \stackrel{id\times m}{\to} & G\times G\\ m\times id\downarrow && \downarrow m \\ G\times G & \stackrel{m}{\to} &G }

(expressing the fact multiplication is associative),

G (1,id) G×G (id,1) = m G×G m G \array{ G & \stackrel{(1,id)}{\to} & G\times G\\ (\id,1)\downarrow &\underset{=}{\searrow}& \downarrow m \\ G\times G & \stackrel{m}{\to} &G }

(telling us that the unit map picks out an element that is a left and right identity), and

G (id,() 1) G×G (() 1,id) 1 m G×G m G \array{ G & \stackrel{(id,(-)^{-1})}{\to} & G\times G\\ ((-)^{-1},id)\downarrow & \underset{1}{\searrow}& \downarrow m \\ G\times G & \stackrel{m}{\to} &G }

(telling us that the inverse map really does take an inverse), where we have let 1:GG1: G \to G denote the composite G*1GG \to * \stackrel{1}{\to} G.

Even if CC doesn't have all binary products, as long as products with GG (and the terminal object **) exist, then one can still speak of a group object GG in CC.

Cogroup

Ring

ring

A (unital, non-commutative) ring is (equivalently)

Here AbAb 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.

Coring

commutative ring object?

Biring

biring

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 RR equipped with ring homomorphisms called coaddition: RRR R \to R \otimes R cozero: R R \to \mathbb{Z} co-additive inverse: RR R \to R comultiplication:RRR R \to R \otimes R and the multiplicative counit: R R \to \mathbb{Z}

satisfying the dual axioms of a commutative ring.

Equivalently, a biring is a commutative ring RR equipped with a lift of the functor

hom(R,):CommRingSet hom(R, -) : CommRing \to Set

to a functor

hom(R,):CommRingCommRing hom(R, -) : CommRing \to CommRing

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 Λ\Lambda, the ring of symmetric polynomials. This is actually a plethysm.

Algebra

Coalgebra

Bialgebra

bialgebra

A bialgebra (or bigebra?) is both an algebra? and a 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?.

More precisely a bialgebra is a monoid? in? the category of coalgebras?. Equivalently, it is a comonoid? in? the category of algebras?. Equivalently, it is a monoid in the category of comonoids in Vect? — or equivalently, a comonoid in the category of monoids in Vect?.

More generally, a bimonoid? in a monoidal category MM is a monoid in the category of comonoids in MM — or equivalently, a comonoid in the category of monoids in MM. So, a bialgebra is a bimonoid in VectVect.

Hopf algebra

Hopf algebra

A kk-bialgebra (A,m,η,Δ,ϵ)(A,m,\eta,\Delta,\epsilon) with multiplication mm, comultiplication Δ\Delta, unit η:kA\eta: k\to A and counit ϵ:Ak\epsilon:A\to k is a Hopf algebra if there exists a kk-linear map

S:AAS : A \to A

called the antipode or coinverse such that m(idS)Δ=m(Sid)Δ=ηϵm\circ(\mathrm{id}\otimes S)\circ \Delta = m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon (as a map AAA\to A). 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 S(1)1=1S(1)1=1, therefore S(1)=1S(1)=1. By linearity of SS this implies that Sηϵ=ηϵS\circ\eta\circ\epsilon = \eta\circ\epsilon.

Revision on July 20, 2012 at 14:22:58 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.