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MONOIDAL FUNCTORS AND EXACT SEQUENCES OF GROUPS FOR HOPF QUASIGROUPS

  • Alvarez, Jose N. Alonso (Departamento de Matematicas Universidad de Vigo Campus Universitario Lagoas-Marcosende) ;
  • Vilaboa, Jose M. Fernandez (Departamento de Matematicas Universidad de Santiago de Compostela) ;
  • Rodriguez, Ramon Gonzalez (Departamento de Matematica Aplicada II Universidad de Vigo Campus Universitario Lagoas-Marcosende)
  • Received : 2020.02.10
  • Accepted : 2020.10.12
  • Published : 2021.03.01

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 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 [3]. 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(HMod) ≅ Pic(C)⊕G(H∗) where Pic(HMod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H∗) the group of group-like morphisms of the dual of H.