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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?

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    $\begingroup$ As far as i can understand, the category of finite dimensional, commutative and cocommutative Hopf algebras over an algebraically closed field of char 0, is equivalent to the category of finite abelian groups (as a consequence of applying the Cartier-Konstant-Milnor-Moore classification theorem of cocommutative hopf algebras in the fin dim case). Does this answer your question ? $\endgroup$ Apr 3 at 22:07
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    $\begingroup$ @KonstantinosKanakoglou Unfortunately, I need to allow infinite-dimensional Hopf algebras too. $\endgroup$ Apr 4 at 14:45

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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:

  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. 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).

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  • $\begingroup$ It occurs to me that from what you said in your question and from the introduction to the paper you linked, it's unclear in what generality the category of commutative and cocommutative Hopf algebras over $k$ is abelian. Of course, any restrictions on $k$ required for that to be true should be applied also in the above Corollary. $\endgroup$
    – Tim Campion
    May 11 at 3:04
  • $\begingroup$ Do you have a more down-to-earth reference for what a locally finitely-presentable category is? I'm finding it really hard to read the nLab article you referenced. $\endgroup$ May 23 at 15:12

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