Galois functors and entwining structures

JOURNAL OF ALGEBRA

卷:324

Article

Mesablishvili, Bachuki^2,3  Wisbauer, Robert¹ 

[1] Dept Math HHU, D-40225 Dusseldorf, Germany

[2] Razmadze Math Inst, GE-0193 Tbilisi, Georgia

[3] Tbilisi Ctr Math Sci, GE-0168 Tbilisi, Georgia

关键词: Corings;    (Galois) comodules;    Galois functors;    Relative injectives (projectives);    Equivalence of categories;   

DOI  :  10.1016/j.jalgebra.2010.04.004

来源: Elsevier

【 摘 要 】

Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a grouplike natural transformation g: I -> G generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to opmonoidal monads with antipode on autonomous monoidal categories (named Hopf monads by Bruguieres and Virelizier) which can be understood as an entwining of two related functors. As well known, for any set G the product G x - defines an endo-functor on the category of sets and this is a Hopf monad if and only if G allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to Galois objects in the sense of Chase and Sweedler. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free