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Several weeks ago I asked this at MathStackExchange, and to my surprise nobody answered.

Recently I understood that I know almost nothing about the category $\operatorname{HopfAlg}$ of Hopf algebras (over a given field $k$). Can anybody enlighten me? I wonder, in particular,

how far does the analogy between Hopf algebras and groups go?

Is, for example, $\operatorname{HopfAlg}$ a category with kernels and cokernels? Or is it possible to assign to each algebra $A$ a natural Hopf algebra $H$ that can be considered as an analogue of "group of automorphisms" of $A$?

This is strange, I can't find any mentionings about this.

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    $\begingroup$ You might enjoy the work of Hans Porst. $\endgroup$
    – Tyrone
    Commented Jan 29, 2021 at 22:17
  • $\begingroup$ @Tyrone thank you, I'll look at them. $\endgroup$
    – Sergei Akbarov
    Commented Jan 29, 2021 at 22:21
  • $\begingroup$ I would have to dig a while to get a perfect reference for the second question (assigning a Hopf algebra to each algebra), so I didn't include it in my answer below. There is a construction of this form that I believe is due to Chelsea Walton. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jan 30, 2021 at 1:17

3 Answers 3

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The category of Hopf algebras has both kernels and cokernels, and indeed it has all limits and colimits. In fact, quite remarkably, it is locally presentable. This is remarkable because the "easy" proofs of local presentability for a category of BLAH BLAH algebras require that BLAH BLAH algebras have only many-to-one operations, whereas Hopf algebras have also one-to-many operations. A good reference (mentioned by Tyrone in the comments) is: Hans-E. Porst. On categories of monoids, comonoids, and bimonoids. Quaest. Math., 31(2):127–139, 2008.

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  • $\begingroup$ Theo, this result about kernels and cokernels, is it contained in another paper? $\endgroup$
    – Sergei Akbarov
    Commented Jan 30, 2021 at 7:48
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    $\begingroup$ (Co)kernels are special cases of (co)limits $\endgroup$
    – Max Horn
    Commented Jan 31, 2021 at 10:52
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The analogy between cocommutative Hopf algebras and groups is remarkably close. The category of (graded) cocommutative $k$-coalgebras is cartesian monoidal under the tensor product and the groups in that category are the cocommutative Hopf algebras. This is discussed in some detail in the paper "Enriched model categories in equivariant contexts" (\url{https://www-intlpress-com.proxy.uchicago.edu/site/pub/pages/journals/items/hha/content/vols/0021/0001/a010/} or \url{https://arxiv.org/pdf/1307.4488.pdf}), by Bertrand Guillou, Jonathan Rubin, and myself. We call these guys ``Hopf groups'' and try to justify the name.

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Concerning the second question, if the algebra A is finite dimensional, a universal bialgebra analogous to End(A) is easily constructed, by considering a suitable quotient of the tensor algebra on the coalgebra C:= (End(A))^*. And, for a bialgebra, the there is also a classical Hopf envelope construction. I've written some details in https://arxiv.org/abs/2008.09937

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