Example 10.3.0.7. The axioms of Definition 10.3.0.6 tend to be satisfied by any category $\operatorname{\mathcal{C}}$ whose objects can be described as “sets with algebraic structure.” For example:

The category of sets is regular.

The category of groups is regular.

The category of abelian groups is regular.

The category of associative rings is regular.

The category of commutative rings is regular.