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I am trying to verify whether the category of bialgebras and then the category of weak bialgebras are cocomplete.

We know that algebraic categories are cocomplete (Thm. 4.5 of this book), so I have been trying to show that these two categories are algebraic.

I suppose that this is something known for bialgebras at least, but I am not able to find any reference about that (or a proof for the matter). Does anyone know a reference or a proof?


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up vote 9 down vote accepted

The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras. (See the edit below for a reference.) Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

We can say more: any category that is algebraic over a locally presentable category is also locally presentable. It follows that the category of bialgebras and the category of Hopf algebras are also cocomplete.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

Edit: I am not sure whether Hopf algebras are coalgebraic over the category of algebras; this is perhaps a difficult problem. The status of this and related universal properties (including the case of bialgebras, which is easier) are well-presented in a paper by Porst. (Link fixed)

Edit: The category of weak bialgebras is also complete and cocomplete; this is proved by similar methods from the theory of accessible categories. Namely, the category of weak bialgebras (like that of bialgebras and of Hopf algebras) can be constructed as an equifier between two natural transformations in the 2-category of accessible categories, just as in the bialgebra case. I'll refer you to another paper of Porst for details (section 2).

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Thanks for the answer! what about the category of weak bialgebras? – Steve B Sep 28 '12 at 17:15
I've just made another edit to answer the question. The paper by Porst doesn't treat weak bialgebras explicitly, but the exact same method applies to them. – Todd Trimble Sep 28 '12 at 19:33
Actually, you should probably take a look at Porst's website: because he has a lot of papers on this sort of topic. – Todd Trimble Sep 28 '12 at 19:36
I've seen in Adamek - Rosicky that an equifer is accessible (Lem. 2.76), so how do we get cocompleteness? – Steve B Oct 19 '12 at 22:17
Well, the underlying functor from weak bialgebras (I guess it's weak bialgebras you're asking about?) to algebras preserves and reflects colimits (in other words, colimits of weak bialgebras are just computed as they would be at the underlying algebra level -- you can check this directly), so cocompleteness of weak bialgebras follows from cocompleteness of algebras. The same argument can be applied towards bialgebras and Hopf algebras. – Todd Trimble Oct 19 '12 at 23:40

The category of bialgebras (as well as the category of Hopf algebras) over a field is complete and cocomplete. Completeness is proved in the paper

A.L. Agore, Limits of coalgebras, bialgebras and Hopf algebras, Proc. Amer. Math. Soc. 139 (2011), 855-863.

Also right at the beginning of the paper the author states (with references):

The categories of coalgebras, bialgebras or Hopf algebras have arbitrary coproducts and coequalizers ([references]), hence these categories are cocomplete.

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