# 41. Group and cogroup objects

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. π

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.)

Comments and questions?