---
date: 2021-10-23 18:22
modification date: Saturday 23rd October 2021 18:22:35
title: cogroup
aliases: [cogroup]

---

Tags: #todo  

# cogroup

> Reference: [One of Qiaochu Yuan's Blog Posts](https://qchu.wordpress.com/2011/01/21/structures-on-hom-sets/)

- Group object: have structure maps
\[
m: G^{\times 2} &\to G \\
\eps: 1 &\to G \\
i: G &\to G
.\]
where 1 is a terminal (?) object.


- Cogroup objects: have structure maps
$$
m: H &\to H^{\times 2} \\
\eps: H &\to 0 \\
i: H &\to H
$$
where $0$ is an initial object.

- Example: $S^n \in \ho\Top$.

- **Importance**: What structure does $H$ need to have such that $\hom(H,\wait)$ has a group structure when applied? The answer is that $H$ is a group object in $\mathcal{C}\op$, or equivalently that $H$ is a cogroup object in $\mathcal{C}$.

- The forgetful functor $U: \Set\to\Grp$ is representable by $\hom_{\Grp}(\ZZ, \wait)$, and the coproduct in $\Grp$ is the free product.

- Recall that $\CRing\op \cong \Sch(\Aff)$, the category of affine schemes.

- The adjoint to the forgetful functor $\CRing \to \Set$ is the free commutative ring on $X$, i.e. $\ZZ[X]$, and is thus representable. The forgetful functor $\CRing \to \Ab$ given by sending a ring to its underlying abelian group is also representable, namely by $\hom_{\Ring}(\ZZ[x], \wait)$. The coproduct in $\Ring$ is the tensor product over $\ZZ$, and the initial object is $\ZZ$.

- $\ZZ[x]$ with its cogroup structure defines the structure of an affine group scheme on $\spec \ZZ[x]$, which represents the "additive group" functor and is called the additive group scheme $\GG_a$. Dualizing, an affine group scheme in the category $\CRing$ is precisely a Hopf algebra.

- Similarly, the forgetful functor $\CRing \to \Ab$ given by sending $R$ to $R\units$ is representable by 
\[
\Hom_{\Ring}(\ZZ[x, x\inv], \wait)
\] 
and the corresponding affine group scheme $\spec \ZZ[x, x \inv]$ is the multiplicative group scheme $\GG_m$.

- Note: the functor $\Sch(\Aff) \to \Set$ sending a ring to its set of prime ideals is not representable (and doesn't preserve products), but the functor 
\[
\Hom_{\Sch(\Aff)}(\spec k, \wait)
\]
sending a scheme to its $k\dash$points for any $k$ is representable (and preserves all limits).