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On the third cohomology group of commutative monoids

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Abstract

We interpret Grillet’s symmetric third cohomology classes of commutative monoids in terms of strictly symmetric monoidal abelian groupoids. We state and prove a classification result that generalizes the well-known one for strictly commutative Picard categories by Deligne, Fröhlich and Wall, and Sinh.

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References

  1. Barr, M., Beck, J.: Homology and standard constructions. In: Seminar on Triples and Categorical Homology. Lecture Notes in Mathematics, vol. 80, pp. 245–335. Springer, Berlin (1969). Reprints in Theory and Applications of Categories, No. 18 (2008)

  2. Borceux, F., Bourn, D.: Mal’cev, Protomodular, Homological and Semi-Abelian Categories, vol. 566. Kluwer Academic Publishers, Philip Drive Norwell (2004)

    Book  MATH  Google Scholar 

  3. Brandt, H.: Über die Axiome des Gruppoids, Vierteljscher. Naturforsch. Ges. Zurich 85, 95–104 (1940)

    MATH  Google Scholar 

  4. Calvo, M., Cegarra, A.M., Quang, N.T.: Higher cohomologies of modules. Algebraic Geom. Topol. 12, 343–413 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calvo, M., Cegarra, A.M., Heredia, B.A.: Structure and classification of monoidal groupoids. Semigroup Forum 87, 35–79 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cegarra, A.M., Garca-Calcines, J.M., Ortega, J.A.: On graded categorical groups and equivariant group extensions. Can. J. Math. 54, 970–997 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cegarra, A.M., Khadmaladze, E.: Homotopy classification of braided graded categorical groups. J. Pure Appl. Algebra 209, 411–437 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cegarra, A.M., Khadmaladze, E.: Homotopy classification of graded Picard categories. Adv. Math. 213, 644–686 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deligne, P.: La formule de dualité globale, exposé XVIII of Théorie des Topos et Cohomologie Etale des Schémas (SGA 4), In: Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1973)

  10. Fröhlich, A., Wall, C.T.C.: Graded monoidal categories. Compos. Math. 28, 229–285 (1974)

    MathSciNet  MATH  Google Scholar 

  11. Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory, Progress in Mathematics. Birkhauser, Boston (1999)

    Book  MATH  Google Scholar 

  12. Grillet, P.A.: Commutative semigroup cohomology. Semigroup Forum 43, 247–252 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grillet, P.A.: Commutative semigroup cohomology. Comm. Alg. 23, 3573–3587 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grillet, P.A.: Cocycles in commutative semigroup cohomology. Comm. Alg. 25, 3427–3462 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grillet, P.A.: Commutative Semigroups (Advances in Mathematics). Kluwer Academic Publisher, Philip Drive Norwell (2001)

    Book  Google Scholar 

  16. Higgins, P.J.: Categories and Groupoids. Van Nostrand Reinhold, New York (1971). Reprints in Theory and Applications of Categories. No. 7 (2005)

  17. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 82, 20–78 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Kurdiani, R., Pirashvili, T.: Functor homology and homology of commutative monoids. Semigroup Forum (2015). doi:10.1007/s00233-014-9683-z

  19. Kelly, G.M.: On Mac Lane’s conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1, 397–402 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leech, J.: Two Papers: \(\cal H\)-Coextensions of Monoids and the Structure of a Band of Groups. AMS, Providence (1975)

    MATH  Google Scholar 

  21. Mac Lane, S.: Cohomology theory of abelian groups. In: Proceedings of the International Congress of Mathematicians, vol. 2, pp. 8–14 (1950)

  22. Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Studies 49, 28–46 (1963)

    MathSciNet  MATH  Google Scholar 

  23. Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  24. Saavedra, N.: Catégories Tannakiennes. Lecture Notes in Mathamatics. Springer, Berlin (1972)

    Google Scholar 

  25. Sinh, H.X.: Gr-catégories. Thèse de Doctorat, Université Paris VII (1975)

  26. Sinh, H.X.: Catégories de Picard restreintes. Acta Math. Vietnamica 7, 117–122 (1982)

    MATH  Google Scholar 

  27. Wells, C.: Extension theories for monoids. Semigroup Forum 16, 13–35 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are much indebted to the referee, whose useful observations greatly improved our exposition. This work has been supported by ‘Dirección General de Investigación’ of Spain, Project: MTM2011-22554, and for the first and third authors also by FPU Grants FPU12-01112 and AP2010-3521, respectively.

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Authors and Affiliations

  1. Department of Algebra, University of Granada, 18071, Granada, Spain

    M. Calvo-Cervera, A. M. Cegarra & B. A. Heredia

Authors
  1. M. Calvo-Cervera
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  2. A. M. Cegarra
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  3. B. A. Heredia
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Correspondence to A. M. Cegarra.

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Communicated by Mark V. Lawson.

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Calvo-Cervera, M., Cegarra, A.M. & Heredia, B.A. On the third cohomology group of commutative monoids. Semigroup Forum 92, 511–533 (2016). https://doi.org/10.1007/s00233-015-9696-2

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