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Pivotal objects in monoidal categories and their Hopf monads

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Aryan Ghobadi

A pair of objects (P,Q) in a monoidal category C, is called a pivotal pair if there exist a family of duality morphisms, making Q both a left dual and a right dual of P. We introduce the correct notion of morphisms between such pairs, and thereby define the pivotal cover of a monoidal category. Given such a pair (P,Q), we construct the category C(P,Q), of objects which intertwine with P and Q in a compatible manner and show that C(P,Q) lifts the monoidal structure of C as well as the closed structure of C, when C is closed. If C has suitable colimits, we construct a family of Hopf monads which correspond to such pairs in C and present the resulting families of braided Hopf algebras and Hopf algebroids, when C is a braided category or the category of bimodules over a base algebra, respectively.

Keywords:
Monoidal category, closed category, pivotal category, Hopf monad, Hopf algebra, tensor category

2020 MSC:
18M05, 18D15, 16T99, 18C20

*Theory and Applications of Categories,*
Vol. 37, 2021,
No. 11, pp 287-325.

Published 2021-03-17.

http://www.tac.mta.ca/tac/volumes/37/11/37-11.pdf

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