Monoidal functors and exact sequences of groups for Hopf quasigroups
J. Korean Math. Soc.
Published online December 3, 2020
José Nicanor Alonso Álvarez, José Manuel Fernández Vilaboa, and Ramón González Rodríguez
University of Vigo, University of Santiago de Compostela
Abstract : In this paper we introduce the notion of strong Galois $H$-progenerator object for a finite cocommutative Hopf quasigroup $H$ in a symmetric monoidal category ${\sf C}$. We prove that the set of isomorphism classes of strong Galois $H$-progenerator objects is a subgroup of the group of strong Galois $H$-objects introduced in \cite{JKMS}. Moreover, we show that strong Galois $H$-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if $H$ is finite, we find exact sequences of Picard groups related with invertible left $H$-(quasi)modules and an isomorphism $Pic(_{{\sf H}}{\sf Mod})\cong Pic({\sf C})\oplus G(H^{\ast})$ where $Pic(_{{\sf H}}{\sf Mod})$ is the Picard group of the category of left $H$-modules, $Pic({\sf C})$ the Picard group of ${\sf C}$, and $G(H^{\ast})$ the group of group-like morphisms of the dual of $H$.
Keywords : Monoidal category, monoidal functor, Hopf (co)quasigroup, (strong) Galois object, Galois group, group-like element, invertible object, Picard group.
MSC numbers : 18D10, 17A01, 20N05, 16T05.
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