2009/09/30 by Bachuki Mesablishvili, Mesablishvili, Bachuki, Robert Wisbauer +1
Mathematics · #16T15 #18A40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.CT #math.QA #msc:16T15 #msc:18A40
paper · pdf · doi:10.48550/arxiv.0909.5590
arxiv created 2009/09/30 · openalex publication_date 2009/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
\em Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of \em 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 \em 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 comonoidal functors on monoidal categories in the sense of A. Bruguières and A. Virelizier. As well-know, for any set G the product G×- defines an endofunctor 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 \em Galois objects in the sense of Chase and Sweedler.