×

Monoidal structures on the categories of quadratic data. (English) Zbl 1472.18016

This paper gives several (co)lax 2-monoidal structures on categories of quadratic data, and several constructions of (co)operads are detailed, recovering important examples from the literature. In Section 2, a definition is given for “quadratic data”, which is essentially the presentation data of an associative algebra. There are also symmetric and skew-symmetric quadratic data, which present commutative and Lie algebras respectively. Several monoidal structures are defined on each of these categories, as well as several monoidal functors relating them and the categories of algebras they present, including the universal enveloping algebra functor, and Koszul duality functors. In Section 3, they give the required interchange law maps to assemble some of these monoidal structures into lax 2-monoidal structures. A lax 2-monoidal structure on a category is a generalization of duoidal categories which allows the interchange law \(\phi_{A,A',B,B'} \colon (A \otimes A') \boxtimes (B \otimes B') \to (A \boxtimes B) \otimes (A' \boxtimes B')\) to not necessarily be an isomorphism. In Section 4, they introduce the category of binary operadic quadratic data as well as multiple (co)lax 2-monoidal structures on it. Section 5 discusses how to obtain quadratic data from topological operads. This is done by applying the Magnus construction to the fundamental group. In this way, they are able to construct many operads of interest, such as the genus zero quantum cohomology operad and the operad classiyfing Gerstenhaber algebras.

MSC:

18M05 Monoidal categories, symmetric monoidal categories
18M50 Bimonoidal, skew-monoidal, duoidal categories
18M60 Operads (general)
18M70 Algebraic operads, cooperads, and Koszul duality
16S37 Quadratic and Koszul algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29, 2010. zbl 1209.18002; MR2724388 · Zbl 1209.18002
[2] Johan Alm and Dan Petersen, Brown’s dihedral moduli space and freedom of the gravity operad, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 50 (2017), no. 5, 1081-1122. DOI 10.24033/asens.2640; zbl 1401.14139; MR3720025; arxiv 1509.09274 · Zbl 1401.14139 · doi:10.24033/asens.2640
[3] V. I. Arnol’d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227-231. DOI 10.1007/BF01098313; zbl 0277.55002; MR0242196 · Zbl 0277.55002 · doi:10.1007/BF01098313
[4] C. Balteanu, Z. Fiedorowicz, R. Schw\"anzl, and R. Vogt, Iterated monoidal categories, Adv. Math. 176 (2003), no. 2, 277-349. DOI 10.1016/S0001-8708(03)00065-3; zbl 1030.18006; MR1982884; arxiv math/9808082 · Zbl 1030.18006 · doi:10.1016/S0001-8708(03)00065-3
[5] Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423-472. DOI 10.1016/0040-9383(95)93237-2; zbl 0898.57001; MR1318886 · Zbl 0898.57001 · doi:10.1016/0040-9383(95)93237-2
[6] Clemens Berger, Op\'erades cellulaires et espaces de lacets it\'er\'es, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 4, 1125-1157. DOI 10.5802/aif.1543; zbl 0853.55007; MR1415960 · Zbl 0853.55007 · doi:10.5802/aif.1543
[7] Alexander Berglund, Koszul spaces, Trans. Amer. Math. Soc. 366 (2014), no. 9, 4551-4569. DOI 10.1090/S0002-9947-2014-05935-7; zbl 1301.55006; MR3217692; arxiv math/1107.0685 · Zbl 1301.55006 · doi:10.1090/S0002-9947-2014-05935-7
[8] Mikhail Bershtein, Vladimir Dotsenko, and Anton Khoroshkin, Quadratic algebras related to the bi-Hamiltonian operad, Int. Math. Res. Not. IMRN (2007), no. 24, Art. ID rnm122, 30 pp. DOI 10.1093/imrn/rnm122; zbl 1149.18003; MR2377009; arxiv math/0607289 · Zbl 1149.18003 · doi:10.1093/i
[9] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, vol. 347, Springer-Verlag, Berlin, 1973. zbl 0285.55012; MR0420609 · Zbl 0285.55012
[10] Francis C. S. Brown, Multiple zeta values and periods of moduli spaces \(\overline{\mathfrak{M}_{0,n}} \), Ann. Sci. \'Ec. Norm. Sup\'er. (4) 42 (2009), no. 3, 371-489. DOI 10.24033/asens.2099; zbl 1216.11079; MR2543329; arxiv math/0606419 · Zbl 1216.11079 · doi:10.24033/asens.2099
[11] U. Buijs, Y. F\'elix, A. Murillo, and D. Tanr\'e, Lie models of simplicial sets and representability of the Quillen functor, Isr. J. Math. 238 (2020), no. 1, 313-358. DOI 10.1007/s11856-020-2026-8; zbl 07247257; MR4145802; arxiv 1508.01442 · Zbl 1460.55011 · doi:10.1007/s11856-020-2026-8
[12] Kuo-tsai Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217-246. DOI 10.2307/1970846; zbl 0227.58003; MR0380859 · Zbl 0227.58003 · doi:10.2307/1970846
[13] Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833-1910. DOI 10.1215/00127094-3120274; zbl 1339.55004; MR3357185; arxiv 1204.4533 · Zbl 1339.55004 · doi:10.1215/00127094-3120274
[14] J. Cirici and G. Horel, Mixed Hodge structures and formality of symmetric monoidal functors, to appear in Ann. Sci. \'Ec. Norm. Sup\'er, 2017. arxiv 1703.06816
[15] Frederick R. Cohen, The homology of \(\mathcal{C}_{n+1} \)-spaces, The homology of iterated loop spaces (Cohen, Frederick R. and Lada, Thomas J. and May, J. Peter), Lecture Notes in Mathematics, vol. 533, Springer-Verlag, Berlin, 1976, pp. vii+490. zbl 0334.55009; MR0436146 · Zbl 0334.55009
[16] Vladimir Dotsenko, Sergei Shadrin, and Bruno Vallette, Toric varieties of Loday’s associahedra and noncommutative cohomological field theories, J. Topol. 12 (2019), no. 2, 463-535. DOI 10.1112/topo.12091; zbl 1421.18002; MR4072173; arxiv 1510.03261 · Zbl 1421.18002 · doi:10.1112/topo.12091
[17] Vladimir Dotsenko, Sergei Shadrin, and Bruno Vallette, The twisting procedure, to appear in the London Mathematical Society Lecture Note Series (Cambridge University Press), 2018. arxiv 1810.02941
[18] V. G. Drinfel’d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \(\rm{Gal}(\overline{\bf Q}/\bf Q)\), Algebra i Analiz 2 (1990), no. 4, 149-181. zbl 0728.16021; MR1080203 · Zbl 0728.16021
[19] Cl\'ement Dupont and Bruno Vallette, Brown’s moduli spaces of curves and the gravity operad, Geom. Topol. 21 (2017), no. 5, 2811-2850. DOI 10.2140/gt.2017.21.2811; zbl 1398.14033; MR3687108; arxiv 1509.08840 · Zbl 1398.14033 · doi:10.2140/gt.2017.21.2811
[20] Pavel Etingof, Andr\'e Henriques, Joel Kamnitzer, and Eric M. Rains, The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points, Ann. of Math. (2) 171 (2010), no. 2, 731-777. DOI 10.4007/annals.2010.171.731; zbl 1206.14051; MR2630055; arxiv math/0507514 · Zbl 1206.14051 · doi:10.4007/annals.2010.171.731
[21] Benoit Fresse, Homotopy of operads and Grothendieck-Teichm\"uller groups, Mathematical Surveys and Monographs, vol. 217 & 218, American Mathematical Society, Providence, RI, 2017. DOI 10.1090/surv/217.1; DOI 10.1090/surv/217.2; zbl 1373.55014(vol.1); zbl 1375.55007(vol.2); MR3616816(vol.2) · Zbl 1373.55014 · doi:10.1090/surv/217.1
[22] Benoit Fresse, Thomas Willwacher, The intrinsic formality of \(E\_n\)-operads, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 7, 2047-2133. DOI 10.4171/JEMS/961; zbl 07227731; MR4107503; arxiv 1503.08699 · Zbl 1445.18014 · doi:10.4171/JEMS/961
[23] E. Getzler, Operads and moduli spaces of genus \(0\) Riemann surfaces, The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkh\"auser Boston, Boston, MA, 1995, pp. 199-230. zbl 0851.18005; MR1363058; arxiv alg-geom/9411004 · Zbl 0851.18005
[24] E. Getzler, Lie theory for nilpotent \(L_\infty \)-algebras, Ann. of Math. (2), 170 (2009), no. 1, 271-301. DOI 10.4007/annals.2009.170.271; zbl 1246.17025; MR2521116; arxiv math/0404003 · Zbl 1246.17025 · doi:10.4007/annals.2009.170.271
[25] V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203-272. DOI 10.1215/S0012-7094-94-07608-4; zbl 0855.18006; MR1301191; arxiv 0709.1228 · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4
[26] V. Ginzburg and M. Kapranov, Erratum to: “Koszul duality for operads”, Duke Math. J. 80 (1995), no. 1, 293. DOI 10.1215/S0012-7094-95-08011-9; zbl 0855.18007; MR1360619 · Zbl 0855.18007 · doi:10.1215/S0012-7094-95-08011-9
[27] F. Guill\'en Santos, V. Navarro, P. Pascual, and A. Roig, Moduli spaces and formal operads, Duke Math. J. 129 (2005), no. 2, 291-335. DOI 10.1215/S0012-7094-05-12924-6; zbl 1120.14018; MR2165544; arxiv math/0402098 · Zbl 1120.14018 · doi:10.1215/S0012-7094-05-12924-6
[28] Phil Hanlon and Michelle Wachs, On Lie \(k\)-algebras, Adv. Math. 113 (1995), no. 2, 206-236. DOI 10.1006/aima.1995.1038; zbl 0844.17001; MR1337108 · Zbl 0844.17001 · doi:10.1006/aima.1995.1038
[29] Andr\'e Henriques and Joel Kamnitzer, Crystals and coboundary categories, Duke Math. J. 132 (2006), no. 2, 191-216. DOI 10.1215/S0012-7094-06-13221-0; zbl 1123.22007; MR2219257; arxiv math/0406478 · Zbl 1123.22007 · doi:10.1215/S0012-7094-06-13221-0
[30] Andr\'e Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20-78. DOI 10.1006/aima.1993.1055; zbl 0817.18007; MR1250465 · Zbl 0817.18007 · doi:10.1006/aima.1993.1055
[31] M. Kapranov and Yu. I. Manin, Modules and Morita theorem for operads, Amer. J. Math. 123 (2001), no. 5, 811-838. DOI 10.1353/ajm.2001.0033; zbl 1001.18004; MR1854112; arxiv math/9906063 · Zbl 1001.18004 · doi:10.1353/ajm.2001.0033
[32] Sean Keel, Intersection theory of moduli space of stable \(n\)-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545-574. DOI 10.2307/2153922; zbl 0768.14002; MR1034665 · Zbl 0768.14002 · doi:10.2307/2153922
[33] Anton Khoroshkin, Quadratic Algebras arising from Hopf operads generated by a single element, Lett. Math. Phys. 110 (2020), no. 8, 2053-2082. DOI 10.1007/s11005-020-01283-z; zbl 07239421; MR4126873; arxiv 1907.0557 · Zbl 1441.18028 · doi:10.1007/s11005-020-01283-z
[34] Toshitake Kohno, S\'erie de Poincar\'e-Koszul associ\'ee aux groupes de tresses pures, Invent. Math. 82 (1985), no. 1, 57-75. DOI 10.1007/BF01394779; zbl 0574.55009; MR0808109 · Zbl 0574.55009 · doi:10.1007/BF01394779
[35] Maxim Kontsevich, Formal (non)commutative symplectic geometry, In The Gel’fand Mathematical Seminars, 1990-1992, pages 173-187. Birkh\"auser Boston, Boston, MA, 1993. zbl 0821.58018; MR1247289 · Zbl 0821.58018
[36] M. Kontsevich and Yu. I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562. DOI 10.1007/BF02101490; zbl 0853.14020; MR1291244; arxiv hep-th/9402147 · Zbl 0853.14020 · doi:10.1007/BF02101490
[37] M. Kontsevich and Yu. I. Manin, Quantum cohomology of a product, Invent. Math. 124 (1996), no. 1-3, 313-339. With an appendix by R. Kaufmann. DOI 10.1007/s002220050055; zbl 0853.14021; MR1369420; arxiv q-alg/9502009 · Zbl 0853.14021 · doi:10.1007/s002220050055
[38] Maxim Kontsevich, Formality conjecture, In Deformation theory and symplectic geometry (Ascona, 1996), volume 20 of Math. Phys. Stud., pages 139-156. Kluwer Acad. Publ., Dordrecht, 1997. zbl 1149.53325; MR1480721 · Zbl 1149.53325
[39] Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35-72, Mosh\'e Flato (1937-1998). DOI 10.1023/A:1007555725247; zbl 0945.18008; MR1718044; arxiv math/9904055 · Zbl 0945.18008 · doi:10.1023/A:1007555725247
[40] Pascal Lambrechts and Ismar Voli\'c, Formality of the little \(N\)-disks operad, Mem. Amer. Math. Soc. 230 (2014), no. 1079, viii+116. DOI 10.1090/memo/1079; zbl 1308.55006; MR3220287; arxiv 0808.0457 · Zbl 1308.55006 · doi:10.1090/memo/1079
[41] Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101-190. DOI 10.24033/asens.1021; zbl 0055.25103; MR0088496 · Zbl 0055.25103 · doi:10.24033/asens.1021
[42] Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer-Verlag, Berlin, 2012. DOI 10.1007/978-3-642-30362-3; zbl 1260.18001; MR2954392 · Zbl 1260.18001 · doi:10.1007/978-3-642-30362-3
[43] S. Mac Lane. Categories for the working mathematician (second edition). Graduate Texts in Mathematics, vol. 5, Springer-Verlag, 1998. zbl 0906.18001; MR1712872 · Zbl 0906.18001
[44] Wilhelm Magnus, \"Uber Beziehungen zwischen h\"oheren Kommutatoren, J. Reine Angew. Math. 177 (1937), 105-115. DOI 10.1515/crll.1937.177.105; zbl 63.0065.01; MR1581549 · JFM 63.0065.01 · doi:10.1515/crll.1937.177.105
[45] Yuri I. Manin, Quantum groups and noncommutative geometry, Universit\'e de Montr\'eal, Centre de Recherches Math\'ematiques, Montreal, QC, 1988. zbl 0724.17006; MR1016381 · Zbl 0724.17006
[46] Yuri I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, vol. 47, American Mathematical Society, Providence, RI, 1999. zbl 0952.14032; MR1702284 · Zbl 0952.14032
[47] Yuri I. Manin, Higher structures, quantum groups, and genus zero modular operad, J. Lond. Math. Soc., II. Ser. 100 (2019), no. 3, 721-730. DOI 10.1112/jlms.12217; zbl 07174667; MR4048717; arxiv 1802.04072 · Zbl 1444.18014 · doi:10.1112/jlms.12217
[48] Martin Markl, Distributive laws and Koszulness, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307-323. DOI 10.5802/aif.1516; zbl 0853.18005; MR1393517 · Zbl 0853.18005 · doi:10.5802/aif.1516
[49] J.P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, vol. 271, Springer-Verlag, Berlin, 1972. zbl 0244.55009; MR0420610 · Zbl 0244.55009
[50] Guy Melan\c{c}on and Christophe Reutenauer, Free Lie superalgebras, trees and chains of partitions, J. Algebraic Combin. 5 (1996), no. 4, 337-351. DOI 10.1023/A:1022400700309; zbl 0871.17003; MR1406458 · Zbl 0871.17003 · doi:10.1023/A:1022400700309
[51] D. Robert-Nicoud, Representing the deformation infinity-groupoid, Algebr. Geom. Topol. 19 (2019), no. 3, 1453-1476. DOI 10.2140/agt.2019.19.1453; zbl 07142603; MR3954288; arxiv 1702.02529 · Zbl 1475.17032 · doi:10.2140/agt.2019.19.1453
[52] D. Robert-Nicoud and B. Vallette, Higher Lie theory, to appear (2020)
[53] Paolo Salvatore and Nathalie Wahl, Framed discs operads and Batalin-Vilkovisky algebras, Q. J. Math. 54 (2003), no. 2, 213-231. DOI 10.1093/qmath/hag012; zbl 1072.55006; MR1989873; arxiv math/0106242 · Zbl 1072.55006 · doi:10.1093/qmath/hag012
[54] Pavol \v Severa and Thomas Willwacher, Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175-206. DOI 10.1215/00127094-1443502; zbl 1241.18008; MR2838354; arxiv 0905.1789 · Zbl 1241.18008 · doi:10.1215/00127094-1443502
[55] Dev P. Sinha, Operads and knot spaces, J. Amer. Math. Soc. 19 (2006), no. 2, 461-486 (electronic). DOI 10.1090/S0894-0347-05-00510-2; zbl 1112.57004; MR2188133; arxiv 0407039 · Zbl 1112.57004 · doi:10.1090/S0894-0347-05-00510-2
[56] Dev P. Sinha, The (non-equivariant) homology of the little disks operad, OPERADS 2009, S\'emin. Congr., vol. 26, Soc. Math. France, Paris, 2013, pp. 253-279. zbl 1277.18012; MR3203375; arxiv math/0610236 · Zbl 1277.18012
[57] Paul Arnaud, Songhafouo Tsopm\'en\'e, Symmetric multiplicative formality of the Kontsevich operad, J. Homotopy Relat. Struct. 13 (2018), no. 1, 225-235. DOI 10.1007/s40062-017-0179-x; zbl 1391.55007; MR3769370; arxiv 1604.08006 · Zbl 1391.55007 · doi:10.1007/s40062-017-0179-x
[58] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes \'Etudes Sci. Publ. Math. 47 (1977), 269-331. DOI 10.1007/BF02684341; zbl 0374.57002; MR0646078 · Zbl 0374.57002 · doi:10.1007/BF02684341
[59] Dmitry E. Tamarkin, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1-2, 65-72. DOI 10.1023/B:MATH.0000017651.12703.a1; zbl 1048.18007; MR2064592; arxiv math/9809164 · Zbl 1048.18007 · doi:10.1023/B:MATH.0000017651.12703.a1
[60] Bruno Vallette, Homology of generalized partition posets, J. Pure Appl. Algebra 208 (2007), no. 2, 699-725. DOI 10.1016/j.jpaa.2006.03.012; zbl 1109.18002; MR2277706; arxiv math/0405312 · Zbl 1109.18002 · doi:10.1016/j.jpaa.2006.03.012
[61] Bruno Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math. 620 (2008), 105-164. DOI 10.1515/CRELLE.2008.051; zbl 1159.18001; MR2427978; arxiv math/0609002 · Zbl 1159.18001 · doi:10.1515/CRELLE.2008.051
[62] Thomas Willwacher, M. Kontsevich’s graph complex and the Grothendieck-Teichm\"uller Lie algebra, Invent. Math., 200 (2015), no. 3, 671-760. DOI 10.1007/s00222-014-0528-x; zbl 1394.17044; MR3348138; arxiv 1009.1654 · Zbl 1394.17044 · doi:10.1007/s00222-014-0528-x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.