Skip to main content
SpringerLink
Log in

Algebras Versus Coalgebras

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of ( F , G )-dimodules associated to two functors \(F,G:\mathbb{A}\to \mathbb{B}\) between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.

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

References

  1. Adámek, J., Porst, H.E.: From varieties of algebras to covarieties of coalgebras. Electron. Notes Theor. Comput. Sci. 44(1) (2001)

  2. Baez, J.C.: R-commutative geometry and quantization of poisson algebras. Adv. Math. 95(1), 61–91 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barr, M.: Composite cotriples and derived functors. In: Sem. Triples Categor. Homology Theory. Springer Lecture Notes in Mathematics, vol. 80, pp. 336–356 (1969)

  4. Barr, M., Wells, Ch.: Toposes, triples and theories. Theory Appl. Categ. 12, 1–288 (2005)

    Google Scholar 

  5. Beck, J.: Triples, modules and cohomology. Ph.D. thesis, Columbia University (1967), republished in: Theory Appl. Categ. 2, 1–59 (2003)

  6. Beck, J.: Distributive laws. In: Eckmann, B. (ed.) Seminar on Triples and Categorical Homology Theory. Springer Lecture Notes in Mathematics, vol. 80, pp. 119–140 (1969)

  7. Bespalov, Y., Drabant, B.: Hopf (bi-)modules and crossed modules in braided monoidal categories. J. Pure Appl. Algebra 123(1–3), 105–129 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras. Part I. Integral theory and C *-structure. J. Algebra 221(2), 385–438 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Borceux, F.: Handbook of Categorical Algebra. Part 2: Categories and structures. In: Encyclopedia of Mathematics and Its Applications, vol. 51. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  10. Borceux, F., Vitale, E.: Azumaya categories. Appl. Categ. Structures 10(5), 449–467 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bruguières, A., Virelizier, A.: Hopf monads. arXiv:math.QA/0604180 (2006)

  12. Brzeziński, T.: Crossed products by a coalgebra. Comm. Algebra 25(11), 3551–3575 (1997)

    Article  MathSciNet  Google Scholar 

  13. Brzeziński, T.: The structure of corings: induction functors, Maschke-type theorem, and Frobenius and Galois-type properties. Algebr. Represent. Theory 5(4), 389–410 (2002)

    Article  MathSciNet  Google Scholar 

  14. Brzeziński, T., Majid, Sh.: Comodule bundles. Comm. Math. Phys. 191(2), 467–492 (1998)

    Article  MathSciNet  Google Scholar 

  15. Brzeziński, T., Nichita, F.F.: Yang-Baxter systems and entwining structures. Comm. Algebra 33(4), 1083–1093 (2005)

    Article  MathSciNet  Google Scholar 

  16. Brzeziński, T., Wisbauer, R.: Corings and Comodules. In: London Mathematical Society Lecture Note Series, vol. 309. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  17. Di Luigi, C., Guccione, J.A., Guccione, J.J.: Brzeziński’s crossed products and braided Hopf crossed products. Comm. Algebra 32(9), 3563–3580 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Caenepeel, S., Ion, B., Militaru, G., Zhu, S.: The factorization problem and the smash biproduct of algebras and coalgebras. Algebr. Represent. Theory 3(1), 19–42 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Caenepeel, S., De Lombaerde, M.: A categorical approach to Turaev’s Hopf group-coalgebras. Comm. Algebra 34(7), 2631–2657 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Caenepeel, S., Wang D., Yin Y.: Yetter-Drinfeld modules over weak Hopf algebras and the center construction. Ann. Univ. Ferrara, Sez. VII Sci. Mat. 51, 69–98 (2005)

    MATH  MathSciNet  Google Scholar 

  21. Eilenberg, S., Moore, J.C.: Adjoint functors and triples. Ill. J. Math. 9, 381–398 (1965)

    MATH  MathSciNet  Google Scholar 

  22. Guccione, J.A., Guccione, J.J.: Theory of braided Hopf crossed products. J. Algebra 261(1), 54–101 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gumm, H.P.: Universelle Coalgebra. In: Allgemeine Algebra Ihringer, Th., Berliner Stud. zur Math., Band, 10, 155–207. Heldermann, Berlin (2003)

    Google Scholar 

  24. Gumm, H.P.: Elements of the general theory of coalgebras. In: LUATCS’99, Rand Africaans University, Johannesburg, South Africa (1999), http://www.mathematik.uni-marburg.de/~gumm/

  25. Hagino, T.: A categorical programming language. Ph.D. thesis, University of Edinburgh (1987)

  26. Hobst, D., Pareigis, B.: Double quantum groups. J. Algebra 242(2), 460–494 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Johnstone, P.T.: Adjoint lifting theorems for categories of modules. Bull. London Math. Soc. 7, 294–297 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  28. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kasangian, S., Lack, S., Vitale, E.M.: Coalgebras, braidings, and distributive laws. Theory Appl. Categ. 13(8), 129–146 (2004)

    MATH  MathSciNet  Google Scholar 

  30. Kharchenko, V.K.: Connected braided Hopf algebras. J. Algebra 307, 24–48 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lack, S., Street, R.: The formal theory of monads II. J. Pure Appl. Algebra 175(1–3), 243–265 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lawvere, F.W.: Functorial semantics of algebraic theories. Proc. Natl. Acad. Sci. 50, 869–872 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lenisa, M., Power, J., Watanabe H.: Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads. Electron Notes Theor. Comput. Sci. 33, 230–260 (2000)

    Article  MathSciNet  Google Scholar 

  34. Loday, J.-L., Dialgebras, Loday, J.-L. (ed.) et al.: Dialgebras and Related Operads. Berlin: Springer. Lecture Notes in Math. 1763, 7–66 (2001)

  35. Mac Lane, S.: Categories for the working mathematician. In: Grad. Texts in Mathematics. Springer New York (1998)

    Google Scholar 

  36. Majid, S.: Algebras and Hopf algebras in braided categories. In: Bergen, J. et al. (ed.) Advances in Hopf Algebras. Lecture Notes Pure and Applied Mathematics, vol. 158, pp. 55–105. Marcel Dekker, New York (1994)

    Google Scholar 

  37. McCrudden, P.: Opmonoidal monads. Theory Appl. Categ. 10, 469–485 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Menini, C., Stefan, D.: Descent theory and Amitsur cohomology of triples. J. Algebra 266(1), 261–304 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Mesablishvili, B.: Monads of effective descent type and comonadicity. Theory Appl. Categ. 16(1), 1–45 (2006), electronic

    MATH  MathSciNet  Google Scholar 

  40. Mesablishvili, B.: Entwining Structures in monoidal Categories, submitted

  41. Moerdijk, I.: Monads on tensor categories. J. Pure Appl. Algebra 168(2,3), 189–208 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  42. Poll, E., Zwangenburg, J.: From Modules and Comodules to Dimodules. University of Nijmegen, The Netherlands http://www.cs.kun.nl/~{erikpoll,janz}

  43. Power, J., Watanabe, H.: Combining a monad and a comonad. Theoret. Comput. Sci. 280(1,2), 137–162 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  44. Schauenburg, P.: On the braiding on a Hopf algebra in a braided category. N.Y. J. Math. 4, 259–263 (1998), electronic

    MATH  MathSciNet  Google Scholar 

  45. Å koda, Z.: Distributive laws for actions of monoidal categories. arXiv:math.CT/0406310 (2004)

  46. Škoda, Z.: Noncommutative localization in noncommutative geometry. In: Ranicki, A. (ed.) Noncommutative Localization in Algebra and Topology, London Mathematical Society Lecture Notes Series, vol. 330. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  47. Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2, 149–168 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  48. Szlachányi, K.: The monoidal Eilenberg-Moore construction and bialgebroids. J. Pure Appl. Algebra 182(2,3), 287–315 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  49. Takeuchi, M.: Survey of braided Hopf algebras. In: Andruskiewitsch, N. et al.(ed.) New Trends in Hopf Algebra Theory, Proc. Coll. Quantum Groups and Hopf Algebras, La Falda, Argentina. American Mathematical Society, Providence, RI (1999); Contemp. Math. 267, 301–323 (2000)

    Google Scholar 

  50. Tanaka, M.: Pseudo-distributive laws and a unified framework for variable binding. Ph.D. thesis, University of Edinburgh (2005)

  51. Tanaka, M., Power, J.: Pseudo-distributive laws and axiomatics for variable binding. High.-Order Symb. Comput. 19(2,3), 305–337 (2006)

    Article  MATH  Google Scholar 

  52. Turi, D., Plotkin, G.: Towards a mathematical operational semantics. In: Proceedings 12th Ann. IEEE Symp. on Logic in Computer Science, LICS’97, Warsaw, Poland (1997)

  53. Van Daele, A., Wang, S.: New braided crossed categories and Drinfeld quantum double for weak Hopf group coalgebras. Comm. Algebra (2007) (in press)

  54. van Osdol, D.H.: Sheaves in regular categories. In: Exact Categories and Categories of Sheaves. Lecture Notes of Mathematics, vol. 236, pp. 223–239. Springer, Berlin (1971)

    Google Scholar 

  55. Wisbauer, R.: Weak corings. J. Algebra 245(1), 123–160 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  56. Wisbauer, R.: On Galois comodules. Comm. Algebra 34(7), 2683–2711 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  57. Wolff, H.: \({\cal V}\)-localizations and \({\cal V}\)-monads. J. Algebra 24, 405–438 (1973)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Düsseldorf, Düsseldorf, Germany

    Robert Wisbauer

Authors
  1. Robert Wisbauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Wisbauer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wisbauer, R. Algebras Versus Coalgebras. Appl Categor Struct 16, 255–295 (2008). https://doi.org/10.1007/s10485-007-9076-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-007-9076-5

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation