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?



Does the category of commutative and cocommutative Hopf algebras have enough injectives?
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](https://ncatlab.org/nlab/show/locally+presentable+category):


1. The category of coalgebras over $k$;
2. The category of cocommutative coalgebras over $k$;
3. The category of algebra objects in either (1) or (2);
4. The category of commutative algebra objects in (1) or (2);
5. 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](https://planetmath.org/fundamentaltheoremofcoalgebras). 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).


