Skip to main content
SpringerLink
Log in

Structure and classification of monoidal groupoids

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech’s cohomology groups of monoids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We thank the referee for this observation.

References

  1. Baez, J.C., Lauda, A.D.: Higher dimensional algebra V: 2-groups. Theory Appl. Categ. 12, 423–492 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Barr, M., Beck, J.: Homology and standard constructions. In: Seminar on Triples and Categorical Homology. Lecture Notes in Math., vol. 80, pp. 245–335. Springer, Berlin (1969)

    Chapter  Google Scholar 

  3. Baues, H.-J., Dreckmann, W.: The cohomology of homotopy categories and the general linear group. K-Theory 3, 307–338 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baues, H.-J., Wirsching, G.: Cohomology of small categories. J. Pure Appl. Algebra 38, 187–211 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beck, J.: Triples, algebras and cohomology. Ph. Dissertation, Columbia University (1967). Reprinted in Theory Appl. Categ. 2, 1–59 (2003)

  6. Breen, L.: Theorie de Schreier superieure. Ann. Sci. Éc. Norm. Super. 25, 465–514 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Brown, R., Spencer, C.B.: G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proc. K. Ned. Akad. Wet. 79, 296–302 (1976)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Carrasco, P., Cegarra, A.M.: Schreier theory for central extensions of categorical groups. Commun. Algebra 24, 4059–4112 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cegarra, A.M., Aznar, E.: An exact sequence in the first variable for torsor cohomology: the 2-dimensional theory of obstructions. J. Pure Appl. Algebra 39, 197–250 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cegarra, A.M., Bullejos, M., Garzón, A.R.: Higher dimensional obstruction theory in algebraic categories. J. Pure Appl. Algebra 49, 43–102 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Cegarra, A.M., Garzón, A.R.: Homotopy classification of categorical torsors. Appl. Categ. Struct. 9, 465–496 (2001)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Deligne, P.: Les immeubles des groupes de tresses généralisés. Invent. Math. 17, 273–302 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duskin, J.: Simplicial Methods and the Interpretation of Triple Cohomology. Mem. Am. Math. Soc., vol. 163 (1975)

    Google Scholar 

  18. Fröhlich, A., Wall, C.T.C.: Equivariant Brauer groups. Contemp. Math. 272, 57–71 (2000)

    Article  Google Scholar 

  19. Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  20. Glenn, P.: Realization of cohomology classes in arbitrary exact categories. J. Pure Appl. Algebra 25, 33–105 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grothendieck, A.: Catégories Fibrées et Déscente, (SGA I, expos, VI). Lecture Notes in Math., vol. 224, pp. 145–194. Springer, Berlin (1971)

    Google Scholar 

  22. Illusie, L.: Complex Cotangent et Déformations II. Lecture Notes in Math., vol. 283. Springer, Berlin (1972)

    Google Scholar 

  23. Inassaridze, H.: Non-Abelian Homological Algebra and Its Applications. Kluwer Academic, Dordrecht (1997)

    Book  MATH  Google Scholar 

  24. Leech, J.: Two Papers: \(\mathcal{H}\)-Coextensions of Monoids and the Structure of a Band of Groups. In Mem. Am. Math. Soc., vol. 157 (1975)

    Google Scholar 

  25. Leech, J.: Extending groups by monoids. J. Algebra 74, 1–19 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Joyal, A.: Foncteurs analytiques et espèces de structures. In: Combinatoire Énumérative, Montréal, Québec, 1985. Lecture Notes in Math., vol. 1234, pp. 55–112. Springer, Berlin (1991)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  29. Pirashvili, T.: Models for the homotopy theory and cohomology of small categories. Soobshch. Akad. Nauk Gruz. 129, 261–264 (1988)

    MathSciNet  MATH  Google Scholar 

  30. Pirashvili, T.: Cohomology of small categories in homotopical algebra. In: K-Theory and Homological Algebra, Tbilisi, 1987–1988. Lecture Notes in Math., vol. 1437, pp. 268–302. Springer, Berlin (1990)

    Chapter  Google Scholar 

  31. Rédei, L.: Die Verallgemeinerung der Schreierschen Erweiterungstheorie. Acta Sci. Math. 14, 252–273 (1952)

    MATH  Google Scholar 

  32. Roos, J.E.: Sur les foncteurs dérivés de \(\varprojlim\). C.R. Acad. Sci. Paris 252, 3702–3704 (1961)

    MathSciNet  MATH  Google Scholar 

  33. Saavedra, N.: Catégories Tannakiennes. Lecture Notes in Math., vol. 265. Springer, Berlin (1972)

    MATH  Google Scholar 

  34. Schreier, O.: Uber die Erweiterung von Gruppen 1. Monatshefte Math. Phys. 34, 165–180 (1926)

    Article  MathSciNet  MATH  Google Scholar 

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

  36. Stasheff, J.: Homotopy associativity of H-spaces I, II. Trans. Am. Math. Soc. 108, 275–312 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  37. Street, R.: Categorical structures. In: Handbook of Algebra, vol. 1, pp. 529–577. North-Holland, Amsterdam (1996)

    Google Scholar 

  38. Ulbrich, K.-U.: Group cohomology for Picard categories. J. Algebra 91, 464–498 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Watts, Ch.E.: A homology theory for small categories. In: Proc. Conf. Categorical Algebra, La Jolla, 1965, pp. 331–336. Springer, New York (1966)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Wells, C.: Extension theories for categories. Preliminary report (1979). Revised version (2001) available from http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf

Download references

Acknowledgements

The authors want to express their gratitud to the anonymous referee, whose helpful comments and remarks improved our exposition. This work has been supported by ‘Dirección General de Investigación’ of Spain, Project: MTM2011-22554, and for the third author also by Ministerio de Educación, Cultura y Deportes of Spain: FPU grant AP2010-3521.

Author information

Authors and Affiliations

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

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

Authors
  1. M. Calvo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Cegarra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. A. Heredia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Cegarra.

Additional information

Communicated by Mark V. Lawson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calvo, M., Cegarra, A.M. & Heredia, B.A. Structure and classification of monoidal groupoids. Semigroup Forum 87, 35–79 (2013). https://doi.org/10.1007/s00233-013-9470-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9470-2

Keywords

Search

Navigation