Monoidal functors and exact sequences of groups for Hopf quasigroups
J. Korean Math. Soc. 2021 Vol. 58, No. 2, 351-381
https://doi.org/10.4134/JKMS.j200069
Published online December 3, 2020
Printed March 1, 2021
Jos\'{e} N. Alonso \'{A}lvarez, Jos\'{e} M. Fern\'{a}ndez Vilaboa, Ram\'{o}n Gonz\'{a}lez Rodr\'{\i}guez
Campus Universitario Lagoas-Marcosende; Universidad de Santiago de Compostela; Campus Universitario Lagoas-Mar\-co\-sen\-de
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 : Primary 18D10, 17A01, 20N05, 16T05
Supported by : The authors were supported by Ministerio de Econom\'{\i}a, Industria y Competitividad of Spain. Agencia Estatal de Investigaci\'on. Uni\'on Europea - Fondo Europeo de Desarrollo Regional. Grant MTM2016-79661-P: Homolog\'{\i}a, homotop\'{\i}a e invariantes categ\'oricos en grupos y \'algebras no asociativas
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