On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$ -algebra
Tóm tắt
In this paper, we develop the
$$A_\infty $$
-analog of the Maurer-Cartan simplicial set associated to an
$$L_\infty $$
-algebra and show how we can use this to study the deformation theory of
$$\infty $$
-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of
$$A_\infty $$
-algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved)
$$A_\infty $$
-algebras to simplicial sets, which sends a complete curved
$$A_\infty $$
-algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of
$$\infty $$
-morphisms of algebras over non-symmetric operads.
Tài liệu tham khảo
Behrend, K., Getzler, E.: Geometric higher groupoids and categories. In: Geometry, analysis and probability, volume 310 of Progr. Math., pp. 1–45. Birkhäuser/Springer, Cham (2017)
Berger, C., Moerdijk, I.: Axiomatic homotopy theory for operads. Comment. Math. Helv. 78(4), 805–831 (2003)
Chuang, J., Holstein, J., Lazarev, A.: Maurer-Cartan moduli and theorems of Riemann-Hilbert type. (2018). arXiv:1802.02549v1
Dolgushev, V.A.: A proof of Tsygan’s formality conjecture for an arbitrary smooth manifold. PhD thesis, MIT (2005)
Dolgushev, V.A., Hoffnung, A.E., Rogers, C.L.: What do homotopy algebras form? Adv. Math. 274, 562–605 (2015)
Dolgushev, V.A., Rogers, C.L.: A version of the Goldman-Millson theorem for filtered \(L_\infty \)-algebras. J. Algebra 430, 260–302 (2015)
Dotsenko, V., Shadrin, S., Vallette, B.: Pre-Lie deformation theory. Mosc. Math. J. 16(3), 505–543 (2016)
Dotsenko, V., Shadrin, S., Vallette, B.: The twisting procedure. (2018). arXiv:1810.02941
Drummond-Cole, G.C., Hirsh, J.: Model structures for coalgebras. Proc. Am. Math. Soc. 144(4), 1467–1481 (2016)
Edward, B.: Curtis. Simplicial homotopy theory. Adv. Math. 6(107–209), 1971 (1971)
Erler, T.: Relating Berkovits and \({A}_\infty \) superstring field theories; small hilbert space perspective. J. High Energy Phys. 2015(10), 157 (2015)
Fresse, B.: Homotopy of operads and Grothendieck-Teichmüller groups. In: Part 1, volume 217 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI. The algebraic theory and its topological background (2017)
Fukaya, K.: Deformation theory, homological algebra and mirror symmetry. In: Geometry and physics of branes (Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., pp. 121–209. IOP, Bristol (2003)
Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory: anomaly and obstruction. In: Part I, volume 46 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence; International Press, Somerville (2009)
Getzler, E.: Lie theory for nilpotent \(L_\infty \)-algebras. Ann. Math. (2) 170(1), 271–301 (2009)
Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory. In: Modern Birkhäuser Classics. Birkhäuser Verlag, Basel (2009) (Reprint of the 1999 edition [MR1711612])
Hamilton, A., Lazarev, A.: Characteristic classes of \(A_\infty \)-algebras. J. Homotopy Relat. Struct. 3(1), 65–111 (2008)
Harper, J.E.: Homotopy theory of modules over operads and non-\(\Sigma \) operads in monoidal model categories. J. Pure Appl. Algebra 214(8), 1407–1434 (2010)
Hinich, V.: Descent of Deligne groupoids. Internat. Math. Res. Not. 5, 223–239 (1997)
Jacob, L.: Higher algebra. (2020). http://www.math.harvard.edu/~lurie/papers/HA.pdf
Keller, B.: Introduction to \(A\)-infinity algebras and modules. Homol. Homotopy Appl. 3(1), 1–35 (2001)
Loday, J.-L., Vallette, B.: Algebraic operads. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346. Springer, Heidelberg (2012)
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)
Nicolás, P.: The bar derived category of a curved dg algebra. J. Pure Appl. Algebra 212(12), 2633–2659 (2008)
Pridham, J.P.: Unifying derived deformation theories. Adv. Math. 224(3), 772–826 (2010)
Robert-Nicoud, D., Wierstra, F.: Convolution algebras and the deformation theory of infinity-morphisms. Homol. Homotopy Appl. 21(1), 351–373 (2019)
Robert-Nicoud, D., Wierstra, F.: Homotopy morphisms between convolution homotopy Lie algebras. J. Noncommut. Geom. 13(4), 1463–1520 (2019)
Stasheff, J.D.: Homotopy associativity of \(H\)-spaces. I, II. Trans. Am. Math. Soc. 108, 275–292 (1963)
Stasheff, J.D.: Homotopy associativity of \(H\)-spaces I, II. Trans. Am. Math. Soc. 108, 293–312 (1963)
Stasheff, J.D.: Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras. In: Quantum groups (Leningrad, 1990), volume 1510 of Lecture Notes in Math., pp. 120–137. Springer, Berlin (1992)
Vallette, B.: Homotopy theory of homotopy algebras. Ann. Inst. Fourier (Grenoble) 70(2), 683–738 (2020)
Wierstra, F.: Algebraic Hopf invariants and rational models for mapping spaces. J. Homotopy Relat. Struct. 14(3), 719–747 (2019)