Lifting Theorems for Tensor Functors on Module Categories

Abstract

Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalised to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory.The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focussing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang-Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

Semientwining Structures and Their Applications

“…Let A be an R-algebra and let B be an R-module. Using our terminology (given in Remark 2.6) and the results of [25], we conclude that the category of semientwining structures over A is isomorphic to the category of lifting of functors from the category of R-modules to the category of right A-modules. Remark 3.8.…”

“…This situation is reviewed in [24]: the semi-entwining case is dealt with in general in item 3.3 (which is transfered from [14]); how this general case is translated to our situation is clear from the discussion in item 5.8 of [24]. This is also presented in subsection 3.1 of [25], where the axioms of semi-entwining structures are given by formula (3.1). We give a general definition of liftings of functors.…”