Applied Categorical Structures

, Volume 19, Issue 5, pp 821–858 | Cite as

The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context

  • Tomasz Brzeziński
  • Adrian Vazquez Marquez
  • Joost VercruysseEmail author


The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived.


Monad Adjoint functor Morita context Eilenberg-Moore category 

Mathematics Subject Classification (2000)



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  1. 1.
    Barr, M., Wells, C.: Toposes, triples and theories. Repr. Theory Appl. Categ. 12, 1–287 (2005)MathSciNetGoogle Scholar
  2. 2.
    Beck, J.M.: Triples, algebras and cohomology, PhD Thesis, Columbia University, 1967. Repr. Theory Appl. Categ. 2, 1–59 (2003)Google Scholar
  3. 3.
    Böhm, G., Menini, C.: Pre-torsors and Galois comodules over mixed distributive laws. Appl. Categ. Struct. (in press). doi: 10.1007/s10485-008-9185-9
  4. 4.
    Brzeziński, T., Vercruysse, J.: Bimodule herds. J. Algebra 321, 2670–2704 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Castaño-Iglesias, F., Gómez-Torrecillas, J.: Wide Morita contexts and equivalences of comodule categories. J. Pure Appl. Algebra 131, 213–225 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eilenberg, S., Moore, J.C.: Adjoint functors and triples. Ill. J. Math. 9, 381–398 (1965)MathSciNetzbMATHGoogle Scholar
  7. 7.
    El Kaoutit, L.: Wide Morita contexts in bicategories. Arab. J. Sci. Eng. 33(2C), 153–173 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Mesablishvili, B.: Monads of effective descent type and comonadicity. Theory Appl. Categ. 16, 1–45 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Tomasz Brzeziński
    • 1
  • Adrian Vazquez Marquez
    • 1
  • Joost Vercruysse
    • 2
    Email author
  1. 1.Department of MathematicsSwansea UniversitySwanseaUK
  2. 2.Faculty of EngineeringVrije Universiteit Brussel (VUB)BrusselsBelgium

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