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• Hopf algebras are special case of cogroups in the category of unital rings.

• In $(hTop_*,\vee)$ (Pointed topological spaces with maps between homotopy classes of pointed continuous maps, and $\vee$ the smash product), the spheres $S^n$ with a fixed points are cogroups. The counit $S^n \rightarrow *$ being the terminal map, the coinverse is the symmetry with the equator $S^{n-1} \subset S^n$ and the comulitplication is $S^n \rightarrow S^n \vee S^n$ is the pinching operator sending $S^n$ to $S^n/S^{n-1} \simeq S^n \vee S^n$. You can moreover see that for $(X,x)$ a pointed topological space, $\operatorname{Hom}_{hTop_*}(S^n,(X,x)) = [S^n,(X,x)]$ is a group. In fact, that's a way to define $\pi_n(X,x)$.

• For coactions maybe you can first see that monoid objects in $\mathbb{Z}-Mod$ with the tensor product, are rings. And (left) actions of rings are left modules over rings. If $G$ is a finite group, I think the function ring $\mathcal{C}(G,\mathbb{R})$ would be a (commutative) Hopf algebra, and in particular a cogroup in $\mathbb{R}-Mod$. And modules that have a coaction on this are exactly representations of $G$.

• Hopf algebras are special case of cogroups in the category of unital rings.

• In $(hTop_*,\vee)$ (Pointed topological spaces with maps between homotopy classes of pointed continuous maps, and $\vee$ the smash product), the spheres $S^n$ with a fixed points are cogroups. The counit $S^n \rightarrow *$ being the terminal map, the coinverse is the symmetry with the equator $S^{n-1} \subset S^n$ and the comulitplication is $S^n \rightarrow S^n \vee S^n$ is the pinching operator sending $S^n$ to $S^n/S^{n-1} \simeq S^n \vee S^n$. You can moreover see that for $(X,x)$ a pointed topological space, $\operatorname{Hom}_{hTop_*}(S^n,(X,x)) = [S^n,(X,x)]$ is a group. In fact, that's a way to define $\pi_n(X,x)$.

• For coactions maybe you can first see that monoid objects in $\mathbb{Z}$-$\textrm{Mod}$ with the tensor product, are rings. And (left) actions of rings are left modules over rings. If $G$ is a finite group, I think the function ring $\mathcal{C}(G,\mathbb{R})$ would be a (commutative) Hopf algebra, and in particular a cogroup in $\mathbb{R}$-$\textrm{Mod}$. And modules that have a coaction on this are exactly representations of $G$.

But I think a better setting would be to fix a unital monoidal category $(\mathcal{C},\otimes,\mathbf{1})$ where you have diagonals (that is, for any $X$, a map $X \rightarrow X \otimes X$ that is natural and compatible with the monoidal category structure, see here for the axioms) and to say that a group object is the data of an object $G$ with maps $G \otimes G \rightarrow G$, $\mathbf{1} \rightarrow G$, and $G\rightarrow G$ called respectively the mulitplication, the unit, the inverse and such that they make some diagram commute (corresponding to the unit, the associativity, and the inverse axioms) as in the nlab page. For instance taking group objects in $(Set,\times,*)$ you recover the classical notion of groups.

But I think a better setting would be to fix a unital monoidal category $(\mathcal{C},\otimes,\mathbf{1})$ where you have diagonals (that is, for any $X$, a map $X \rightarrow X \otimes X$ that is natural and compatible with the monoidal category structure, see here for the axioms. This assumption is important for the inverse axiom because you want to talk about elements of the for $x\otimes x$ for $x\in X$ (and then talk about $i(x)\otimes x$). In a general monoidal category the existence of such diagonal map is not automatic) and to say that a group object is the data of an object $G$ with maps $G \otimes G \rightarrow G$, $\mathbf{1} \rightarrow G$, and $G\rightarrow G$ called respectively the mulitplication, the unit, the inverse and such that they make some diagram commute (corresponding to the unit, the associativity, and the inverse axioms) as in the nlab page. For instance taking group objects in $(Set,\times,*)$ you recover the classical notion of groups.

But I think a better setting would be to fix a unital monoidal category $(\mathcal{C},\otimes,\mathbf{1})$ and to say that a group object is the data of an object $G$ with maps $G \otimes G \rightarrow G$, $\mathbf{1} \rightarrow G$, and $G\rightarrow G$ called respectively the mulitplication, the unit, the inverse and such that they make some diagram commute (corresponding to the unit, the associativity, and the inverse axioms) as in the nlab page. For instance taking group objects in $(Set,\times,*)$ you recover the classical notion of groups.

Once this settled up, you can say that a cogroup object in $(\mathcal{C},\otimes,\mathbf{1})$ is a group object in $(\mathcal{C}^{\operatorname{op}},\otimes,\mathbf{1})$ with the same monoidal structure, and a cogroup action is a group action in $(\mathcal{C}^{\operatorname{op}},\otimes,\mathbf{1})$. This does not fit in the first approach of cogroup I have given but it is more general because in that setting you can talk about groups/cogroups for monoidal structures other that the cartesians one. You can write what it is explicitely by taking the axioms of a group object and taking the opposite of all arrows.

But I think a better setting would be to fix a unital monoidal category $(\mathcal{C},\otimes,\mathbf{1})$ where you have diagonals (that is, for any $X$, a map $X \rightarrow X \otimes X$ that is natural and compatible with the monoidal category structure, see here for the axioms) and to say that a group object is the data of an object $G$ with maps $G \otimes G \rightarrow G$, $\mathbf{1} \rightarrow G$, and $G\rightarrow G$ called respectively the mulitplication, the unit, the inverse and such that they make some diagram commute (corresponding to the unit, the associativity, and the inverse axioms) as in the nlab page. For instance taking group objects in $(Set,\times,*)$ you recover the classical notion of groups.

Once this settled up, you can say that a cogroup object in $(\mathcal{C},\otimes,\mathbf{1})$ is a group object in $(\mathcal{C}^{\operatorname{op}},\otimes,\mathbf{1})$ with the same monoidal structure, but here you need to assume you have codiagonals (maps $X \otimes X \rightarrow X$), and a cogroup action is a group action in $(\mathcal{C}^{\operatorname{op}},\otimes,\mathbf{1})$. This does not fit in the first approach of cogroup I have given but it is more general because in that setting you can talk about groups/cogroups for monoidal structures other that the cartesians one. You can write what it is explicitely by taking the axioms of a group object and taking the opposite of all arrows.