Coming from having an interest in topos theory, it might be interesting to add the property of a category having a natural numbers object. This is an initial object in the category of T-objects, where T is the variety with a nullary operation and a unary operation and no identities.
Some examples: any Grothendieck topos has a natural numbers object, and so does any finitary algebraic category; FinSet does not. The category of topological spaces has one (the discrete topology on N), and so does the category of compact Hausdorff spaces ($\beta \mathbb{N}$).
Coming from having an interest in topos theory, it might be interesting to add the property of a category having a natural numbers object. This is an initial object in the category of T-objects, where T is the variety with a nullary operation and a unary operation and no identities.
Some examples: any Grothendieck topos has a natural numbers object, and so does any finitary algebraic category; FinSet does not. The category of topological spaces has one (the discrete topology on N), and so does the category of compact Hausdorff spaces ($\beta \mathbb{N}$ ).