# Contents * table of contents {:toc} ## Monoid [[nLab:monoid]] A monoid in a monoidal category $C$ is defined to be an object $M$ equipped with a multiplication $\mu: M \otimes M \to M$ and a unit $\eta: I \to M$ satisfying the **associative law**: ![A pic](http://upload.wikimedia.org/wikipedia/commons/b/b1/Monoid_mult.png) and the **left and right unit laws**: ![A pic](http://upload.wikimedia.org/wikipedia/commons/1/10/Monoid_unit.png) Here $\alpha$ is the [[nLab:associator]] in $C$, while $\lambda$ and $\rho$ are the left and right [[nLab:unitor|unitors]]. Classical monoids are of course just monoids in [[nLab:Set]] with the [[nLab:cartesian product]]. ### Comonoid ### Bimonoid ## Group [[nLab:group object]] A **group object** or **internal group** in a category $C$ with binary [[nLab:product]]s and a [[nLab:terminal object]] $*$ is an object $G$ in $C$ and arrows $$ 1:* \to G $$ (the unit map) $$ (-)^{-1}:G\to G $$ (the inverse map) and $$ m:G\times G \to G $$ (the multiplication map), such that the following diagrams commute: $$ \array{ G\times G\times G & \stackrel{id\times m}{\to} & G\times G\\ m\times id\downarrow && \downarrow m \\ G\times G & \stackrel{m}{\to} &G } $$ (expressing the fact multiplication is associative), $$ \array{ G & \stackrel{(1,id)}{\to} & G\times G\\ (\id,1)\downarrow &\underset{=}{\searrow}& \downarrow m \\ G\times G & \stackrel{m}{\to} &G } $$ (telling us that the unit map picks out an element that is a left and right identity), and $$ \array{ G & \stackrel{(id,(-)^{-1})}{\to} & G\times G\\ ((-)^{-1},id)\downarrow & \underset{1}{\searrow}& \downarrow m \\ G\times G & \stackrel{m}{\to} &G } $$ (telling us that the inverse map really does take an inverse), where we have let $1: G \to G$ denote the composite $G \to * \stackrel{1}{\to} G$. Even if $C$ doesn\'t have *all* binary products, as long as products with $G$ (and the terminal object $*$) exist, then one can still speak of a group object $G$ in $C$. ### Cogroup ## Ring [[nLab:ring]] A (unital, non-commutative) **ring** is (equivalently) * a [[nLab:monoid]] [[nLab:internalization|internal to]] [[nLab:Ab]]. * a [[nLab:enriched category|category enriched over]] [[nLab:Ab]] with one object. * a [[nLab:ringoid]] with one object. Here $Ab$ is the category of [[nLab:abelian group|abelian groups]], made into a monoidal category using the tensor product of abelian groups. A **commutative** (unital) ring is an [[nLab:abelian monoid]] object in [[nLab:Ab]]. ### Coring [[nLab:commutative ring object]] ### Biring [[nLab:biring]] A biring is defined to be a [[nLab:internalization|commutative ring object]] in the [[nLab:opposite category|opposite]] of the category of [[nLab:commutative ring|commutative rings]] (also known as the category of [[nLab:affine schemes]]). A biring is hence a ring $R$ equipped with ring [[nLab:homomorphisms]] called **coaddition**: $ R \to R \otimes R $ **cozero**: $ R \to \mathbb{Z} $ **co-additive inverse**: $ R \to R $ **comultiplication**:$ R \to R \otimes R $ and the **multiplicative counit**: $ R \to \mathbb{Z} $ satisfying the dual axioms of a commutative ring. Equivalently, a biring is a commutative ring $R$ equipped with a lift of the functor $$ hom(R, -) : CommRing \to Set $$ to a functor $$ hom(R, -) : CommRing \to CommRing $$ Birings form a [[nLab:monoidal category]] thanks to the fact that functors of this form are closed under composition. A [[nLab:monoid object]] in this monoidal category is called a [[nLab:plethory]]. A plethory is an example of a [[nLab:Tall–Wraith monoid]]. The most important example of a biring is $\Lambda$, the ring of [[nLab:symmetric polynomials]]. This is actually a [[nLab:plethysm]]. ## Algebra ### Coalgebra ### Bialgebra [[nLab:bialgebra]] A bialgebra (or bi[[gebra]]) 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]] [[internalization|in]] the category of [[coalgebra|coalgebras]]. Equivalently, it is a [[comonoid]] [[internalization|in]] the category of [[algebra|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 $M$ is a monoid in the category of comonoids in $M$ --- or equivalently, a comonoid in the category of monoids in $M$. So, a bialgebra is a bimonoid in $Vect$. ## Hopf algebra [[nLab:Hopf algebra]] A $k$-[[nLab:bialgebra]] $(A,m,\eta,\Delta,\epsilon)$ with multiplication $m$, comultiplication $\Delta$, unit $\eta: k\to A$ and counit $\epsilon:A\to k$ is a **Hopf algebra** if there exists a $k$-linear map $$S : A \to A$$ called the **antipode** or **coinverse** such that $m\circ(\mathrm{id}\otimes S)\circ \Delta = m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon$ (as a map $A\to A$). 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 $S(1)1=1$, therefore $S(1)=1$. By linearity of $S$ this implies that $S\circ\eta\circ\epsilon = \eta\circ\epsilon$.