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Semigroup Forum

, Volume 99, Issue 1, pp 67–105 | Cite as

Cohomology of monoids with operators

  • A. M. CegarraEmail author
RESEARCH ARTICLE
  • 88 Downloads

Abstract

This paper is dedicated to introducing and studying a cohomology theory for monoids enriched with the action by endomorphisms of a fixed monoid of operators. This equivariant cohomology theory extends both Whitehead’s cohomology for groups with operators and Leech’s cohomology for monoids.

Keywords

Monoid with operators Equivariant extension Equivariant cohomology 

Notes

References

  1. 1.
    Baez, J.C., Dolan, J.: Categorification. arXiv:math/9802029 (1998)
  2. 2.
    Baez, J.C., Lauda, A.D.: Higher-dimensional Algebra V: 2-groups. Theory Appl. Categ. 12, 433–491 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baues, H.-J., Dreckmann, W.: The cohomology of homotopy categories and the general linear group. K-theory 3, 307–338 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baues, H.-J., Jibladze, M.: Classification of abelian track categories. K-theory 25, 299–311 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cegarra, A.M., Garía-Calcines, J.M., Ortega, J.A.: Cohomology of groups with operators. Homol. Homotopy Appl. 4, 1–23 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cegarra, A.M., Garía-Calcines, J.M., Ortega, J.A.: On graded categorical groups and equivariant group extensions. Can. J. Math. 54, 970–997 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calvo, M., Cegarra, A.M., Heredia, B.-A.: Structure and classification of monoidal groupoids. Semigroup Forum 87, 35–79 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cockcroft, W.H.: Interpretation of vector cohomology groups. Am. J. Math. 76, 599–619 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grillet, P.A.: Left coset extensions. Semigroup Forum 7, 200–263 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grillet, P.A.: Semigroups: An Introduction to the Structure Theory, vol. 193. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  11. 11.
    Higgins, P.J.: Categories and groupoids. Repr. Theory Appl. Categ. 7, 1–178 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Leech, J.: \({\cal{H}} \)-coextensions of monoids. Mem. Am. Math. Soc. 157, 1–66 (1975)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Leech, J.: The cohomology of monoids. Unpublished lecture notes (1976)Google Scholar
  15. 15.
    Leech, J.: Cohomology theory for monoids congruences. Houston J. Math. 11, 207–223 (1985)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mac Lane, S.: Categories for the Working Mathematician. GTM 5, 2nd edn. Springer, BerlinGoogle Scholar
  17. 17.
    Noohi, B.: Group cohomology with coefficients in a crossed module. J. Inst. Math. Jussieu 10, 359–404 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Roos, J.E.: Sur les foncteurs dérivés de \(\varprojlim \). Appl. C. R. Acad. Sci. Paris 252, 3702–3704 (1961)zbMATHGoogle Scholar
  19. 19.
    Saavedra, N.: Catégories Tannakiennes, vol. 265. Springer, Berlin (1972). Lecture Notes in MathzbMATHGoogle Scholar
  20. 20.
    Street, R.: Categorical Structures, from: Handbook of Algebra, vol. 1, pp. 529–577. North-Holland, Amsterdam (1996)zbMATHGoogle Scholar
  21. 21.
    Watts, C.E.: A homology theory for small categories. In: Proc. Conf. Categorical Algebra, La Jolla 1965, pp. 331–336. Springer, New York (1966)Google Scholar
  22. 22.
    Wells, C.: Extension theories for monoids. Semigroup Forum 16, 13–35 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Whitehead, J.H.C.: On Group extensions with operators. Q. J. Math. Oxf. 2, 219–228 (1950)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of AlgebraUniversity of GranadaGranadaSpain

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