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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
Aryan Ghobadi. Pivotal objects in monoidal categories and their Hopf monads. Theory and applications of categories, Tome 37 (2021), pp. 287-325. http://geodesic.mathdoc.fr/item/TAC_2021_37_a10/
@article{TAC_2021_37_a10,
author = {Aryan Ghobadi},
title = {Pivotal objects in monoidal categories and their {Hopf} monads},
journal = {Theory and applications of categories},
pages = {287--325},
year = {2021},
volume = {37},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2021_37_a10/}
}