Monads and comonads on module categories

Gabriella Böhm, Tomasz Brzeziński, Robert Wisbauer

Research output: Contribution to journalArticle

29 Citations (Scopus)


Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor - ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor - ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of - ⊗A B and comodules (or coalgebras) of - ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA (B, -) and HomA (C, -) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA (B, -)-comodules is isomorphic to the category of B-modules, while the category of HomA (C, -)-modules (called C-contramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and HomA (C, -)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of HomA (C, -)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor HomR (H, -) and the category of mixed HomR (H, -)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H* is a Hopf algebra.

Original languageEnglish
Pages (from-to)1719-1747
Number of pages29
JournalJournal of Algebra
Issue number5
Publication statusPublished - Sep 1 2009


  • (Co)modules
  • (Co)monads
  • (Co)rings
  • Contramodules
  • Hopf algebras
  • Hopf monads

ASJC Scopus subject areas

  • Algebra and Number Theory

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