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Bihom hopf algebras viewed as hopf monoids

par Böhm, Gabriella;Vercruysse, Joost
Référence Contemporary mathematics - American Mathematical Society, 771, page (1-41)
Publication Publié, 2021-01-01

Article révisé par les pairs

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Résumé :

We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them Lax+Oplax0-monoidal. Dually, we consider Lax0Oplax+-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define Lax+0Oplax0+-duoidal categories with compatible Lax+Oplax0- and Lax0Oplax+-monoidal structures. We introduce comonoids in Lax+Oplax0-monoidal categories, monoids in Lax0Oplax+-monoidal categories and bimonoids in Lax+0Oplax0+-duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a Lax+0Oplax0+-duoidal category D to any symmetric monoidal category V. The unital BiHom-monoids, counital BiHom-comonoids, and unital and counital BiHom-bimonoids in V, due to Grazianu et al., are identified with the monoids, comonoids and bimonoids in D, respectively.