cogroup

Reference: One of Qiaochu Yuan’s Blog Posts

Group object: have structure maps \begin{align*} m: G^{\times 2} &\to G \\ {\varepsilon}: 1 &\to G \\ i: G &\to G .\end{align*} where 1 is a terminal (?) object.
Cogroup objects: have structure maps \begin{align*} m: H &\to H^{\times 2} \\ {\varepsilon}: H &\to 0 \\ i: H &\to H \end{align*} where \(0\) is an initial object.
Example: \(S^n \in {\mathsf{ho}}{\mathsf{Top}}\).
Importance: What structure does \(H\) need to have such that \(\hom(H,{-})\) has a group structure when applied? The answer is that \(H\) is a group object in \(\mathcal{C}^{\operatorname{op}}\), or equivalently that \(H\) is a cogroup object in \(\mathcal{C}\).
The forgetful functor \(U: {\mathsf{Set}}\to{\mathsf{Grp}}\) is representable by \(\hom_{{\mathsf{Grp}}}({\mathbf{Z}}, {-})\), and the coproduct in \({\mathsf{Grp}}\) is the free product.
Recall that \(\mathsf{CRing}^{\operatorname{op}}\cong {\mathsf{Sch}}({\mathsf{Aff}})\), the category of affine schemes.
The adjoint to the forgetful functor \(\mathsf{CRing}\to {\mathsf{Set}}\) is the free commutative ring on \(X\), i.e. \({\mathbf{Z}}[X]\), and is thus representable. The forgetful functor \(\mathsf{CRing}\to {\mathsf{Ab}}\) given by sending a ring to its underlying abelian group is also representable, namely by \(\hom_{\mathsf{Ring}}({\mathbf{Z}}[x], {-})\). The coproduct in \(\mathsf{Ring}\) is the tensor product over \({\mathbf{Z}}\), and the initial object is \({\mathbf{Z}}\).
\({\mathbf{Z}}[x]\) with its cogroup structure defines the structure of an affine group scheme on \(\operatorname{Spec}{\mathbf{Z}}[x]\), which represents the “additive group” functor and is called the additive group scheme \({\mathbf{G}}_a\). Dualizing, an affine group scheme in the category \(\mathsf{CRing}\) is precisely a Hopf algebra.
Similarly, the forgetful functor \(\mathsf{CRing}\to {\mathsf{Ab}}\) given by sending \(R\) to \(R^{\times}\) is representable by \begin{align*} \mathop{\mathrm{Hom}}_{\mathsf{Ring}}({\mathbf{Z}}[x, x^{-1}], {-}) \end{align*} and the corresponding affine group scheme \(\operatorname{Spec}{\mathbf{Z}}[x, x ^{-1}]\) is the multiplicative group scheme \({\mathbf{G}}_m\).
Note: the functor \({\mathsf{Sch}}({\mathsf{Aff}}) \to {\mathsf{Set}}\) sending a ring to its set of prime ideals is not representable (and doesn’t preserve products), but the functor \begin{align*} \mathop{\mathrm{Hom}}_{{\mathsf{Sch}}({\mathsf{Aff}})}(\operatorname{Spec}k, {-}) \end{align*} sending a scheme to its \(k{\hbox{-}}\)points for any \(k\) is representable (and preserves all limits).