In this lecture we define a group object in a category. The idea is to write the axioms for the definition of a group in terms of asking certain diagrams to commute. The terminology makes sense, since:

• A group object in the category $\mathsf{Sets}$ is (surprise!) a group.
• A group object in the category $\mathsf{Top}$ is a topological group.
• A group object in the category $\mathsf{Groups}$ is an abelian group. 🙃

We then define the dual notion of a cogroup, which is obtained by reversing all the arrows. This terminology is slightly less natural:

• The only cogroup object in $\mathsf{Sets}$ is the empty set.
• A cogroup object in the category $\mathsf{Groups}$ is a free group.

We then prove that an object $G$ in a category $\mathsf{c}$ is a group object if and only if $\mathrm{Hom}(G, \square)$ takes values in $\mathsf{Groups}$ (rather than $\mathsf{Sets}$). Similarly, an object $K$ in a category $\mathsf{C}$ is a group object if and only if $\mathrm{Hom}(\square, K)$ takes values in $\mathsf{Groups}$ (rather than $\mathsf{Sets}$).

Next lecture we will investigate (co)group objects in $\mathsf{hTop}_*$ (the category we are primarily interested in.)