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. πŸ™ƒ

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.)


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