Generalized Operads and Their Inner Cohomomorphisms

  • Dennis V. Borisov
  • Yuri I. Manin
Part of the Progress in Mathematics book series (PM, volume 265)


In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad-like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring-like” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts.


Operads algebras inner cohomomorphisms symmetry and deformations in noncommutative geometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BarW]
    M. Barr and Ch. Wells, Toposes, triples and theories. Grundlehren der mathematischen Wissenschaften 278, Springer Verlag, 1985, 358 pp.Google Scholar
  2. [Bat]
    M. Batanin, The Eckmann-Hilton argument, higher operads and E n-spaces. math.CT/0207281 Sep 2003, 58 pp.Google Scholar
  3. [Bek]
    T. Beke, Sheafifiable homotopy model categories. II. J. Pure Appl. Algebra 164 (2001), no. 3, 307–324.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [Ben]
    J. Bénabou, Introduction to bicategories in 1967 Reports of the Midwest Category Seminar, Springer Verlag, 1–77.Google Scholar
  5. [BeMa]
    K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants. Duke Math. Journ. 85 (1996), no. 1, 1–60.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [BerDW]
    R. Berger, M. Dubois-Violette and M. Wambst, Homogeneous algebras. J. Algebra 261 (2003), 172–185. Preprint math.QA/0203035zbMATHCrossRefMathSciNetGoogle Scholar
  7. [BerM]
    R. Berger and N. Marconnet, Koszul and Gorenstein properties for homogeneous algebras. Algebras and representation theory 9 (2006), 67–97.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [BerMo]
    C. Berger and I. Moerdijk, Resolution of colored operads and rectification of homotopy algebras. math.AT/0512576 (to appear in Cont. Math., vol. in honor of Ross Street).Google Scholar
  9. [Bl]
    D. Blanc, New model categories from old. Journal of Pure and Applied Algebra 109 (1996), 37–60.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Bo1]
    F. Borceux, Handbook of categorical algebra 1. Basic category theory. Encyclopedia of Mathematics and its applications, Cambridge University Press, 1994, 360 pp.Google Scholar
  11. [Bo2]
    F. Borceux, Handbook of categorical algebra 2. Categories and structures. Encyclopedia of Mathematics and its applications, Cambridge University Press, 1994, 360 pp.Google Scholar
  12. [CaGa]
    J.G. Cabello and A.R. Garzón, Closed model structures for algebraic models of n-types. Journal of Pure and Applied Algebra 103 (1995), no. 3, 287–302.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Cra]
    S.E. Crans, Quillen closed model structures for sheaves. Journal of Pure and Applied Algebra 101 (1995), 35–57.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [DeMi]
    P. Deligne and J. Milne, Tannakian categories. In: Hodge cycles, motives and Shimura varieties, Springer Lecture Notes in Math. 900 (1982), 101–228.Google Scholar
  15. [Fr]
    B. Fresse, Koszul duality for operads and homology of partition posets. Contemp. Math. 346 (2004), 115–215.MathSciNetGoogle Scholar
  16. [Ga]
    W.L. Gan, Koszul duality for dioperads. Math. Res. Lett. 10 (2003), no. 1, 109–124.zbMATHMathSciNetGoogle Scholar
  17. [GeKa1]
    E. Getzler and M. Kapranov, Cyclic operads and cyclic homology. In: Geometry, Topology and Physics for Raoul Bott (ed. by S.-T. Yau), International Press 1995, 167–201.Google Scholar
  18. [GeKa2]
    E. Getzler and M. Kapranov, Modular operads. Compositio Math. 110 (1998), no. 1, 65–126.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [GiKa]
    V. Ginzburg and M. Kapranov, Koszul duality for operads. Duke Math. J. 76 (1994), no. 1, 203–272.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [GoMa]
    A. Goncharov and Yu. Manin, Multiple zeta-motives and moduli spaces \( \overline M _{0,n} \) Compos. Math. 140 (2004), no. 1, 1–14. Preprint math.AG/0204102.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [GrM]
    S. Grillo and H. Montani, Twisted internal COHOM objects in the category of quantum spaces. Preprint math.QA/0112233.Google Scholar
  22. [Hin]
    V. Hinich, Homological algebra of homotopy algebras. Communications in Algebra 25 (1997), no. 10, 3291–3323.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [KaMa]
    M. Kapranov and Yu. Manin, Modules and Morita theorem for operads. Am. J. of Math. 123 (2001), no. 5, 811–838. Preprint math.QA/9906063.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [KoMa]
    M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164 (1994), no. 3, 525–562.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [KS]
    G.M. Kelly and R. Street, Review of the elements of 2-categories in Category Seminar, Springer Lecture Notes in Mathematics 420 (1974), 75–103.MathSciNetCrossRefGoogle Scholar
  26. [LoMa]
    A. Losev and Yu. Manin, Extended modular operad. In: Frobenius Manifolds, ed. by C. Hertling and M. Marcolli, Vieweg & Sohn Verlag, Wiesbaden, 2004, 181–211. Preprint math.AG/0301003.Google Scholar
  27. [Ma1]
    Yu. Manin, Some remarks on Koszul algebras and quantum groups. Ann. Inst. Fourier XXXVII (1987), no. 4, 191–205.MathSciNetGoogle Scholar
  28. [Ma2]
    Yu. Manin, Quantum groups and non-commutative geometry. Publ. de CRM, Université de Montréal, 1988, 91 pp.Google Scholar
  29. [Ma3]
    Yu. Manin, Topics in noncommutative geometry. Princeton University Press, 1991, 163 pp.Google Scholar
  30. [Ma4]
    Yu. Manin, Notes on quantum groups and quantum de Rham complexes. Teoreticheskaya i Matematicheskaya Fizika 92 (1992), no. 3, 425–450. Reprinted in Selected papers of Yu.I. Manin, World Scientific, Singapore 1996, 529–554.MathSciNetGoogle Scholar
  31. [Mar]
    M. Markl, Operads and PROPs. Preprint math.AT/0601129.Google Scholar
  32. [MarShSt]
    M. Markl, St. Shnider and J. Stasheff, Operads in Algebra, Topology and Physics. Math. Surveys and Monographs, vol. 96, AMS 2002.Google Scholar
  33. [MarkSh]
    I. Markov and Y. Shi, Simulating quantum computation by contracting tensor network. Preprint quant-ph/0511069.Google Scholar
  34. [Mer]
    S. Merkulov, PROP profile of deformation quantization and graph complexes with loops and wheels. Preprint math.QA/0412257.Google Scholar
  35. [PP]
    A. Polishchuk and L. Positselski, Quadratic algebras. University Lecture series, No. 37, AMS 2005.Google Scholar
  36. [Pow]
    A.J. Power, A 2-categorical pasting theorem. Journ. of Algebra 129 (1990), 439–445.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [Q]
    D. Quillen, Homotopical Algebra. Springer Lecture Notes in Mathematics 43, Berlin, 1967.Google Scholar
  38. [R]
    C. Rezk, Spaces of algebra structures and cohomology of operads. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1996.Google Scholar
  39. [S]
    J. Spaliński, Strong homotopy theory of cyclic sets. Journal of Pure and Applied Algebra 99 (1995), no. 1, 35–52.CrossRefMathSciNetzbMATHGoogle Scholar
  40. [Va1]
    B. Vallette, A Koszul duality for PROPs. Preprint math.AT/0411542 (to appear in the Transactions of the AMS).Google Scholar
  41. [Va2]
    B. Vallette, Manin’s products, Koszul duality, Loday algebras and Deligne conjecture. Preprint math.QA/0609002.Google Scholar
  42. [Va3]
    B. Vallette, Free monoid in monoidal abelian categories. Preprint math.CT/0411543.Google Scholar
  43. [Zo]
    P. Zograf, Tensor networks and the enumeration of regular subgraphs. Preprint math.CO/0605256.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Dennis V. Borisov
    • 1
  • Yuri I. Manin
    • 1
    • 2
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Max-Planck-Institut für MathematikBonnGermany

Personalised recommendations