Does the category of commutative and cocommutative Hopf algebras have enough injectives? It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about projectives?
 A: Over a field $k$, the answer is yes for injectives; I'm not sure about projectives. Over $\mathbb Z$ or other commutative rings, I really don't know -- the use of the fundamental theorem of coalgebra below seems pretty essential (and the fundamental theorem of coalgebra fails over $\mathbb Z$).
Over a field, in fact more is true:
Claim: Let $k$ be a field. Then the following categories are locally finitely presentable:

*

*The category of coalgebras over $k$;


*The category of cocommutative coalgebras over $k$;


*The category of algebra objects in either (1) or (2);


*The category of commutative algebra objects in (1) or (2);


*As in (3) or (4), but with restricting to objects with antipodes.
Moreover, the natural tensor product on each of these categories is a symmetric monoidal structure preserving filtered colimits, with compact unit.
Corollary: The category of commutative, cocommutative Hopf algebras over a field $k$ is a Grothendieck abelian category (and in particular has enough injectives).
Proof: Every locally finitely presentable abelian category is Grothendieck.
Proof of Claim:
The conclusion for (1) and (2) follows from the fundamental theorem of coalgebra. Then (3) and (4) follow: in general if you have a locally finitely-presentable category with a monoidal structure with compact unit and which preserves filtered colimits, its category of monoids will be locally finitely-presentable, and similarly for commutative monoids. Finally, (5) follows because Hopf algebras are closed among bialgebras under filtered colimits (when the monoidal product has compact unit and preserves filtered colimits).
