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Cotwists of Bicomonads and BiHom-bialgebras

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Abstract

This paper presents the definition of the cotwists of a bicomonad, which offers a generalization of cotwists of bimonoids in duoidal categories. We show that a new bicomonad can be deduced from the cotwist, and their corepresentations are monoidal isomorphic. As applications, the cotwists of bialgebroids and of BiHom-bialgebras are discussed.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, People’s Republic of China

    Xiaohui Zhang & Dingguo Wang

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  1. Xiaohui Zhang
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  2. Dingguo Wang
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Correspondence to Xiaohui Zhang.

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Presented by: Michel Brion

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The work was partially supported by the National Natural Science Foundation of China (No. 11801304, 11801306, 11871301), and the Project funded by China Postdoctoral Science Foundation (No. 2018M630768).

Appendix: Some diagrammatic proofs

Appendix: Some diagrammatic proofs

Proof of Lemma 4.3.

⇒: If κ′ is the convolution inverse of κ, then for any \(X,Y \in \mathcal {M}\), we have

Thus \(\overline {\kappa ^{\prime }}\ast \overline {\kappa } = \varepsilon \diamond \varepsilon \). Note that in this diagram, the formulae (A2), (U4), (3.1) and the naturality are used. Similarly, one can check that \(\overline {\kappa }\ast \overline {\kappa ^{\prime }} = \varepsilon \diamond \varepsilon \). □

Proof of Proposition 4.4.

We only need to prove \({\overline {A^{\kappa }}}_{2} = {{\overline {A}}^{\overline {\kappa }}}_{2}\). To prove this, for any \(X,Y \in \mathcal {M}\), we have

This completes our substantiation. □

Proof of Lemma 4.7, verification of ⇒:

Suppose that Diagram (3.2) holds. For any \(X,Y,Z \in \mathcal {M}\), we compute as follows.

On the one hand, we have

Note that in this diagram, Eqs. (A1) and (3.2) are used.

On the other hand, we have

This completes the prove. □

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Zhang, X., Wang, D. Cotwists of Bicomonads and BiHom-bialgebras. Algebr Represent Theor 23, 1355–1385 (2020). https://doi.org/10.1007/s10468-019-09888-2

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