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Let $A$ and $B$ be two Hopf algebras, and denote by $\mathcal{M}^A$ and $\mathcal{M}^B$ their respective categories of right comodules. If we have a monoidal equivalence between $\mathcal{M}^A$ and $\mathcal{M}^B$ then what can we saw about the relationship between $A$ and $B$? What properties do they share in common, and what is an example of two non-isomorphic Hopf algebras satisfying tis property?

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  • $\begingroup$ If $A$ and $B$ are commutative and you ask for a symmetric monoidal equivalence then you’re in the realm of Tannaka-Krein duality for affine group schemes, although I’m not sure what exact hypotheses are necessary to conclude that $A$ and $B$ are isomorphic (I think working over a field suffices?). $\endgroup$
    – Qiaochu Yuan
    Commented Nov 24, 2020 at 23:44

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There have been some classic papers, on the development of a Morita theory for equivalent Categories of comodules over coalgebras. I believe one of the oldest and most complete works, which develops the theory parallel to the theory of equivalent Categories of modules over algebras, is:

  • Morita theorems for Categories of comodules, M. Takeuchi, J. Fac. Sci. Univ. Tokyo, 24, (1977), p. 629-644

(Some older partial results are also cited in the epilogue of the paper).
Lots of the results presented there, have been generalized at a more rescent work, considering equivalences of comodule Categories for coalgebras over rings:

  • Equivalences of comodule categories for coalgebras over rings, K. Al-Takhman, Journal of Pure and Applied Algebra, 173, (2002), p. 245 – 271
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