Journal of Homotopy and Related Structures

, Volume 11, Issue 3, pp 375–407 | Cite as

Maurer–Cartan spaces of filtered \(L_{\infty }\)-algebras

  • Sinan Yalin


We study several homotopical and geometric properties of Maurer–Cartan spaces for \(L_{\infty }\)-algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role especially in the deformation theory of algebraic structures. In particular, we prove that the Maurer–Cartan simplicial set preserves fibrations and quasi-isomorphisms. Then we present an algebraic geometry viewpoint on Maurer–Cartan moduli sets, and we compute the tangent complex of the associated algebraic stack.



I would like to thank Gennaro di Brino and Bertrand Toen for useful discussions about stacks.


  1. 1.
    Atiyah, M.F., MacDonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, London, Don Mills (1969)Google Scholar
  2. 2.
    Berglund, A.: Rational homotopy theory of mapping spaces via Lie theory for \(L_{\infty }\)-algebras. arXiv:1110.6145 (preprint)
  3. 3.
    Borel, A.: Linear algebraic groups. Graduate Texts in Mathematics, vol. 126, 2nd edn. Springer, Berlin (1969)Google Scholar
  4. 4.
    Cartier, P.: Groupes algébriques et groupes formels. In: Colloq. Théorie des Groupes Algébriques (Bruxelles). Librairie Universitaire, Louvain (1962)Google Scholar
  5. 5.
    Chacholski, W., Scherer, J.: Homotopy theory of diagrams. Mem. Amer. Math. Soc. 736, 155 (2002)Google Scholar
  6. 6.
    Deligne, P.: La formule de dualité globale. In: Théorie des topos et cohomologie étale des schémas (SGA 4), exposé XVIII, Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1973)Google Scholar
  7. 7.
    Dolgushev, V.A.: Stable formality quasi-isomorphisms for Hochschild cochains I. arXiv:1109.6031
  8. 8.
    Dolgushev, V.A., Rogers, C.L.: A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras. arXiv:1407.6735
  9. 9.
    Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gerstenhaber, M., Schack, S.: Algebras, bialgebras, quantum groups, and algebraic deformations. Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990). Contemp. Math., vol. 134, pp. 51–92. Amer. Math. Soc., Providence (1992)Google Scholar
  11. 11.
    Getzler, E.: Lie theory for nilpotent \(L_{\infty }\)-algebras. Ann. Math. (2) 170(1), 271–301 (2009)Google Scholar
  12. 12.
    Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory. Progress in Mathematics, vol. 174. Birkhäuser (1999)Google Scholar
  13. 13.
    Goldman, W.M., Millson, J.J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds. Publ. Math. I.H.E.S. 67, 43–96 (1988)Google Scholar
  14. 14.
    Gómez, T.L.: Algebraic stacks. Proc. Indian Acad. Sci. Math. Sci. 111(1), 1–31 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hinich, V.: Homological algebra of homotopy algebras. Commun. Algebra 25(10), 3291–3323 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hovey, M.: Model categories. Mathematical Surveys and Monographs, vol. 63. AMS (1999)Google Scholar
  17. 17.
    Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures. I, II., Ann. Math. 67(2), 328–466 (1958)Google Scholar
  18. 18.
    Laumont, G., Moret-Bailly, L.: Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 39. Springer, Berlin (2000)Google Scholar
  19. 19.
    Loday, J.L., Vallette, B.: Algebraic operads, Grundlehren Math. Wiss., vol. 346. Springer, Heidelberg (2012)Google Scholar
  20. 20.
    Lurie, J.: Moduli problems for ring spectra. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 1099–1125. Hindustan Book Agency, New Delhi (2010)Google Scholar
  21. 21.
    MacLane, S.: Categories for the working mathematician. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (1998)Google Scholar
  22. 22.
    Manetti, M.: Deformation theory via differential graded Lie algebras. Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), pp. 21–48. Scuola Norm. Sup, Pisa (1999)Google Scholar
  23. 23.
    Manetti, M.: Lectures on deformations of complex manifolds (deformations from differential graded viewpoint). Rend. Mat. Appl. (7) 24(1), 1–183 (2004)Google Scholar
  24. 24.
    Markl, M.: Intrinsic brackets and the L-infinity deformation theory of bialgebras. J. Homotopy Relat. Struct. 5(1), 177–212 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Merkulov, S., Vallette, B.: Deformation theory of representations of prop(erad)s. II. J. Reine Angew. Math. 636(2009), 123–174 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Neumann, F.: Algebraic stacks and moduli of vector bundles, IMPA Mathematical Publications 27, 27th Brazilian Mathematics Colloquium (IMPA), Rio de Janeiro (2009)Google Scholar
  27. 27.
    Schlessinger, M.: Functors of Artin rings. Trans. Am. Math. Soc. 130, 208–222 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schlessinger, M., Stasheff, J.: Deformation theory and rational homotopy type. arXiv:1211.1647
  29. 29.
    Schnider, S., Sternberg, S.: Quantum groups. From coalgebras to Drinfeld algebras. A guided tour, Graduate Texts in Mathematical Physics, II. International Press, Cambridge (1993)Google Scholar
  30. 30.
    Sullivan, D.: Infinitesimal computations in topology. Publ. Math. I.H.E.S. 47, 269–331 (1977)Google Scholar
  31. 31.
    Vallette, B.: A Koszul duality for PROPs. Trans. Am. Math. Soc. 359(10), 4865–4943 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Weibel, C.: An introduction to homological algebra. Cambridge studies in advanced mathematics, vol. 38. Cambridge Univ Press (1994)Google Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2015

Authors and Affiliations

  1. 1.Mathematics Research UnitUniversity of LuxembourgLuxembourgLuxembourg

Personalised recommendations