cogroup object in the category of pointed set.

Question

Does the category of pointed set have cogroup objects and if they exist what are they? Can we describe a simple (and non trivial) exemple of a cogroup in that category?

Answer 1

The answer is yes: every cocartesian category admits cogroups. In the case of $\mathbf{Set}^*$, the category of pointed sets, a cogroup is nothing but a set-theoretic cogroup where the comultiplication, counit and inverse maps preserve the point.

In details a cogroup in $\mathbf{Set}^*$ amounts to the following data:

• a pointed set $(X,x)$
• a comultiplication map $m \colon X \to X \vee X$ where $$X \vee X= X \times \{0\} \cup X \times \{1\}/\sim $$ (with $\sim$ being the smallest equivalence relation such that $(x,0)\sim (x,1)$) such that $m(x)$ is the equivalence class of $(x,0)$ and $(x,1)$, that is $$m(x)=[(x,0)]=\{(x,i) \in X \times \{0\} \cup X \times \{1\} \colon (x,i) \sim (x,0)\}$$
• counit is given by the unique morphism $$e \colon X \to \bullet$$ with value in the initial object of $\mathbf{Set}^*$, which is the singleton set having its only element as a chosen point
• the coinverse map is a map $i \colon X \to X$ which such that $i(x)=x$. These data have to satisfy the axioms of a cogroup (which are the dual of groups' axioms).

As for an example of cogroup: in $\mathbf{Set}^*$ you have always the trivial cogroup $(\bullet,m^\bullet,e_\bullet,i_\bullet)$, having as support the singleton set, and whose structural morphisms (comultiplication, counit and coinverse) are given by the only possible morphisms from this set to itself (remind that $\bullet \vee \bullet \cong \bullet$).