Group (Categorical)

A group is a group object in the category $\mathbf{Set}$. That is, a set $G$ equipped with morphisms: \[ m : G \times G \to G, \quad e : 1 \to G, \quad i : G \to G \] such that the following identities hold:

1. Associativity: $m \circ (m \times \mathrm{id}_G) = m \circ (\mathrm{id}_G \times m)$.
2. Identity laws: $m \circ (e \times \mathrm{id}_G) = \mathrm{id}_G$, $\quad m \circ (\mathrm{id}_G \times e) = \mathrm{id}_G$.
3. Inverse laws: $m \circ (i \times \mathrm{id}_G) = e \circ !$, $\quad m \circ (\mathrm{id}_G \times i) = e \circ !$, where $! : G \to 1$ is the unique morphism.

This is equivalent to Definition~Group (Classical), but phrased entirely in terms of morphisms and composition.