Abstract
We construct a symmetric monoidal category \({\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}\) whose objects are shifted L ∞ -algebras equipped with a complete descending filtration. Morphisms of this category are “enhanced” infinity morphisms between shifted L ∞ -algebras. We prove that any category enriched over \({\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}\) can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L ∞ -algebra point of view. We use this construction in a subsequent paper (Dolgushev et al. Adv. Math. 274, 562–605, 2015) to produce a simplicial model of a (∞,1)-category whose objects are homotopy algebras of a fixed type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dolgushev, V.A.: A Proof of Tsygan’s formality conjecture for an arbitrary smooth manifold, Ph.D. thesis, M.I.T., arXiv:math/0504420
Dolgushev, V.A.: Stable formality quasi-isomorphisms for Hochschild cochains, arXiv:1109.6031
Dolgushev, V.A., Hoffnung, A.E., Rogers, C.L.: What do homotopy algebras form? Adv. Math. 274, 562–605 (2015) arXiv:1406.1751
Dolgushev, V.A., Paljug, B.J.: Tamarkin’s construction is equivariant with respect to the action of the Grothendieck-Teichmueller group. JHRS. doi:10.1007/s40062-015-0115-x. arXiv:1402.7356
Dolgushev, V.A., Rogers, C.L.: Notes on Algebraic Operads, Graph Complexes, and Willwacher’s Construction, Mathematical aspects of quantization. Contemp. Math. 583, 25–145 (2012). Amer. Math. Soc., Providence, RI arXiv: 1202.2937
Dolgushev, V.A., Rogers, C.L., Willwacher, T.: Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields. Ann. Math. (2) 182(3), 855–943 (2015) arXiv:1211.4230
Dolgushev, V.A., Tamarkin, D.E., Tsygan, B.L.: The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal. J. Noncommut. Geom. 1 (1), 1–25 (2007) arXiv:math/0605141
Dolgushev, V., Willwacher, T.: Operadic Twisting – with an application to Deligne’s conjecture. J. Pure Appl. Algebra 219(5), 1349–1428 (2015) arXiv:1207.2180
Getzler, E.: Lie theory for nilpotent L ∞ -algebras. Ann. Math. (2) 170(1), 271–301 (2009) arXiv:math/0404003
Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory, vol. 174. Progress in Mathematics, Birkhäuser, Basel (1999)
Hinich, V.: Descent of Deligne groupoids. Int. Math. Res. Notices 5, 223–239 (1997) arXiv:alg-geom/9606010
Kelly, G.M.: Basic concepts of enriched category theory, vol. 64. Cambridge University Press, Cambridge (1982)
Willwacher, T.: M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra. Invent. Math. 200(3), 671–760 (2015) arXiv:1009. 1654
Willwacher, T.: The Homotopy braces formality morphism, to appear in Duke Math. Journal arXiv:1109.3520
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dolgushev, V.A., Rogers, C.L. On an Enhancement of the Category of Shifted L ∞ -Algebras. Appl Categor Struct 25, 489–503 (2017). https://doi.org/10.1007/s10485-016-9424-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9424-4