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Curved Koszul duality of algebras over unital versions of binary operads. (English) Zbl 1508.18017

The theory of Koszul operads gives a reduced construction of cofibrant resolutions in algebra categories. These resolutions can be used to compute derived functors on algebras. The constructions of such resolutions involves two steps. Let \(P\) be the operad that governs our category of algebras. In the first step, we have to compute the Koszul dual cooperad of the operad \(P\) from a presentation of this operad by generators and relations. Then the Koszul duality functors return a natural cofibrant resolution functor on the category of \(P\)-algebras. But we can also compute, in a second step, a reduced cofibrant resolution of \(P\)-algebras when we have good algebras with a presentation of this algebra.
The definition of the Koszul dual of an operad only works for good (Koszul) operads. The Koszul duality theory was originally defined for operads equipped with a presentation with binary generating operations and homogeneous quadratic relations. The author of the paper under review deals with operads \(uP\) obtained by adding a unit operation (a constant) to such a binary quadratic Koszul operad \(P\). The main purpose of paper is precisely to give a construction of reduced resolutions of algebras equipped with quadratic-linear-constant presentation over \(uP\).
The author also explains the application of his construction to the case of symplectic Poisson \(n\)-algebras \(A_{n;D} = \mathbb{R}[x_1,\dots,x_D,\xi_1,\dots,\xi_D]\), which is generated by variables \((x_i,\xi_i)\) such that \(\deg(x_i) = 0\), \(\deg(\xi_i) = n-1\), and which are equipped with a Poisson bracket of degree \(n-1\) satisfying \(\{x_i,\xi_j\} = \delta_{i j}\). This algebra admits a presentation with quadratic-linear-constant relations over the operad \(uPois_n\) that governs unital Poisson \(n\)-algebras. He proves that this algebra is Koszul, and describes its reduced cofibrant resolution. He uses this resolution to compute the factorization homology \(\int_M A_{n;D}\) of a simply connected parallelized closed \(n\)-manifold \(M\) with coefficients in \(A_{n;D}\). (He proves that this factorization homology is one-dimensional.)

MSC:

18M70 Algebraic operads, cooperads, and Koszul duality
16S37 Quadratic and Koszul algebras
18G10 Resolutions; derived functors (category-theoretic aspects)
17B63 Poisson algebras
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[1] Ayala, David; Francis, John, Factorization homology of topological manifolds, J. Topol., 8, 4, 1045-1084 (2015) · Zbl 1350.55009
[2] Beilinson, Alexander; Drinfeld, Vladimir, Chiral Algebras, American Mathematical Society Colloquium Publications, vol. 51 (2004), American Mathematical Society: American Mathematical Society Providence, RI, pp. vi+375 · Zbl 1138.17300
[3] Campos, Ricardo; Idrissi, Najib; Lambrechts, Pascal; Willwacher, Thomas, Configuration spaces of manifolds with boundary (2018), preprint
[4] Chuang, Joseph; Lazarev, Andrey; Mannan, W. H., Cocommutative coalgebras: homotopy theory and Koszul duality, Homol. Homotopy Appl., 18, 2, 303-336 (2016) · Zbl 1362.16036
[5] Costello, Kevin; Gwilliam, Owen, Factorization Algebras in Quantum Field Theory, vol. 1, New Mathematical Monographs, vol. 31 (2017), Cambridge University Press: Cambridge University Press Cambridge, pp. ix+387 · Zbl 1377.81004
[6] Döppenschmitt, Lennart, Factorization homology of polynomial algebras (2018), preprint
[7] Francis, John, The tangent complex and Hochschild cohomology of \(\mathcal{E}_n\)-rings, Compos. Math., 149, 3, 430-480 (2013) · Zbl 1276.18008
[8] Fresse, Benoit, Koszul duality of operads and homology of partition posets, (Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory. Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemp. Math., vol. 346 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 115-215 · Zbl 1077.18007
[9] Fresse, Benoit, Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal, 13, 2, 237-312 (2006) · Zbl 1141.55006
[10] Fresse, Benoit, Modules over Operads and Functors, Lecture Notes in Mathematics, vol. 1967 (2009), Springer-Verlag: Springer-Verlag Berlin, pp. x+308 · Zbl 1178.18007
[11] Fresse, Benoit, Homotopy of Operads and Grothendieck-Teichmüller Groups, vol. 1, Mathematical Surveys and Monographs, vol. 217 (2017), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, pp. xlvi+532 · Zbl 1373.55014
[12] Fresse, Benoit; Willwacher, Thomas, The intrinsic formality of \(E_n\)-operads, J. Eur. Math. Soc., 22, 7, 2047-2133 (2020) · Zbl 1445.18014
[13] Gálvez-Carrillo, Imma; Tonks, Andrew; Vallette, Bruno, Homotopy Batalin-Vilkovisky algebras, J. Noncommut. Geom., 6, 3, 539-602 (2012) · Zbl 1258.18005
[14] Getzler, E., Operads and moduli spaces of genus 0 Riemann surfaces, (The Moduli Space of Curves. The Moduli Space of Curves, Texel Island, 1994. The Moduli Space of Curves. The Moduli Space of Curves, Texel Island, 1994, Progr. Math., vol. 129 (1995), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 199-230 · Zbl 0851.18005
[15] Getzler, E., Resolving mixed Hodge modules on configuration spaces, Duke Math. J., 96, 1, 175-203 (1999) · Zbl 0986.14005
[16] Getzler, Ezra; Jones, J. D.S., Operads, homotopy algebra and iterated integrals for double loop spaces (1994), preprint
[17] Ginzburg, Victor; Kapranov, Mikhail, Koszul duality for operads, Duke Math. J., 76, 1, 203-272 (1994) · Zbl 0855.18006
[18] Hirsh, Joseph; Millès, Joan, Curved Koszul duality theory, Math. Ann., 354, 4, 1465-1520 (2012) · Zbl 1276.18009
[19] Idrissi, Najib, The Lambrechts-Stanley model of configuration spaces, Invent. Math., 216, 1, 1-68 (2019) · Zbl 1422.55031
[20] Knudsen, Ben, Higher enveloping algebras, Geom. Topol., 22, 7, 4013-4066 (2018) · Zbl 1460.17018
[21] Kontsevich, Maxim, Formal (non)commutative symplectic geometry, (The Gel’fand Mathematical Seminars. The Gel’fand Mathematical Seminars, 1990-1992 (1993), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 173-187 · Zbl 0821.58018
[22] Kontsevich, Maxim, Operads and motives in deformation quantization, Lett. Math. Phys., 48, 1, 35-72 (1999) · Zbl 0945.18008
[23] Lambrechts, Pascal; Stanley, Don, Poincaré duality and commutative differential graded algebras, Ann. Sci. Éc. Norm. Supér. (4), 41, 4, 495-509 (2008) · Zbl 1172.13009
[24] Lambrechts, Pascal; Volić, Ismar, Formality of the Little N-Disks Operad, Mem. Amer. Math. Soc., vol. 230(1079) (2014), pp. viii+116 · Zbl 1308.55006
[25] Le Grignou, Brice, Algebraic operads up to homotopy (2017), preprint · Zbl 1364.18006
[26] Le Grignou, Brice, Homotopy theory of unital algebras, Algebraic Geom. Topol., 19, 3, 1541-1618 (2019) · Zbl 1436.18019
[27] Loday, Jean-Louis; Vallette, Bruno, Algebraic Operads, Grundlehren der Mathematischen Wissenschaften, vol. 346 (2012), Springer: Springer Heidelberg, pp. xxiv+634 · Zbl 1260.18001
[28] Lyubashenko, Volodymyr, Homotopy unital \(A_\infty \)-algebras, J. Algebra, 329, 190-212 (2011) · Zbl 1227.18008
[29] Lyubashenko, Volodymyr, Curved homotopy coalgebras, Appl. Categ. Struct., 25, 6, 991-1036 (2017) · Zbl 1388.16034
[30] Markarian, Nikita, Weyl n-algebras, Commun. Math. Phys., 350, 2, 421-442 (2017) · Zbl 1360.58018
[31] Markarian, Nikita; Lee Tanaka, Hiro, Factorization homology in 3-dimensional topology, (Mathematical Aspects of Quantum Field Theories. Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud. (2015), Springer: Springer Switzerland), 213-231 · Zbl 1315.81071
[32] Markl, Martin, Distributive laws and Koszulness, Ann. Inst. Fourier (Grenoble), 46, 2, 307-323 (1996), numdam: AIF_1996__46_2_307_0 · Zbl 0853.18005
[33] Maunder, James, Koszul duality and homotopy theory of curved Lie algebras, Homol. Homotopy Appl., 19, 1, 319-340 (2017) · Zbl 1388.18026
[34] McCleary, John, A User’s Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58, 561 (2001), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0959.55001
[35] Millès, Joan, The Koszul complex is the cotangent complex, Int. Math. Res. Not. IMRN, 3, 607-650 (2012) · Zbl 1247.18007
[36] Petersen, Dan, Minimal models, GT-action and formality of the little disk operad, Sel. Math. New Ser., 20, 3, 817-822 (2014) · Zbl 1312.55008
[37] Polishchuk, Alexander; Positsel’ski, Leonid, Quadratic Algebras, University Lecture Series, vol. 37 (2005), American Mathematical Society: American Mathematical Society Providence, RI, pp. xii+159 · Zbl 1145.16009
[38] Positsel’skii, Leonid E., Nonhomogeneous quadratic duality and curvature, Funkc. Anal. Prilozh., 27, 3, 57-66 (1993), 96 · Zbl 0826.16041
[39] Priddy, Stewart B., Koszul resolutions, Trans. Am. Math. Soc., 152, 39-60 (1970) · Zbl 0261.18016
[40] Quillen, Daniel, Rational homotopy theory, Ann. Math. (2), 90, 205-295 (1969), JSTOR:1970725 · Zbl 0191.53702
[41] Roca i. Lucio, Victor, Curved operadic calculus (2022), prepublished
[42] Sullivan, Dennis, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 47, 47, 269-331 (1977), (1978), numdam: PMIHES_1977__47__269_0 · Zbl 0374.57002
[43] Tamarkin, Dmitry E., Formality of chain operad of little discs, Lett. Math. Phys., 66, 1-2, 65-72 (2003) · Zbl 1048.18007
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