The little bundles operad

Müller, Lukas¹ ; Woike, Lukas²

¹Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom, Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom
²Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 4, pp. 2029-2070

Résumé

Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an aspherical topological operad that we call the little bundles operad. As our main result, we describe this operad as a groupoid-valued operad in terms of generators and relations and prove that the categorical little bundles algebras are precisely braided G–crossed categories in the sense of Turaev. Moreover, we prove that the evaluation on the circle of a homotopical two-dimensional equivariant topological field theory yields a little bundles algebra up to coherent homotopy.

Détail

DOI : 10.2140/agt.2020.20.2029

Classification : 18D50, 18D10, 57R56
Keywords: operad, topological field theory, braid group, monoidal category, braided monoidal category

Bibliographie

[1] D Ben-Zvi, J Francis, D Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909 | DOI

[2] M Benini, A Schenkel, L Woike, Operads for algebraic quantum field theory, preprint (2017)

[3] C Berger, I Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 31 | DOI

[4] J M Boardman, R M Vogt, Homotopy-everything H–spaces, Bull. Amer. Math. Soc. 74 (1968) 1117 | DOI

[5] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973) | DOI

[6] G E Bredon, Topology and geometry, 139, Springer (1993) | DOI

[7] D Calaque, C Scheimbauer, A note on the (∞,n)–category of cobordisms, Algebr. Geom. Topol. 19 (2019) 533 | DOI

[8] A Clebsch, Zur Theorie der Riemann’schen Fläche, Math. Ann. 6 (1873) 216 | DOI

[9] P Deligne, D Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969) 75 | DOI

[10] J S Ellenberg, A Venkatesh, C Westerland, Homological stability for Hurwitz spaces and the Cohen–Lenstra conjecture over function fields, Ann. of Math. 183 (2016) 729 | DOI

[11] P Etingof, D Nikshych, V Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010) 209 | DOI

[12] B Fresse, Homotopy of operads and Grothendieck–Teichmüller groups, I : The algebraic theory and its topological background, 217, Amer. Math. Soc. (2017)

[13] S Galatius, U Tillmann, I Madsen, M Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009) 195 | DOI

[14] C Galindo, Coherence for monoidal G–categories and braided G–crossed categories, J. Algebra 487 (2017) 118 | DOI

[15] S Gelaki, D Naidu, D Nikshych, Centers of graded fusion categories, Algebra Number Theory 3 (2009) 959 | DOI

[16] E Getzler, Batalin–Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994) 265 | DOI

[17] S Hollander, A homotopy theory for stacks, Israel J. Math. 163 (2008) 93 | DOI

[18] A Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891) 1 | DOI

[19] M Kapranov, V Voevodsky, Braided monoidal 2–categories and Manin–Schechtman higher braid groups, J. Pure Appl. Algebra 92 (1994) 241 | DOI

[20] R M Kaufmann, Orbifold Frobenius algebras, cobordisms and monodromies, from: "Orbifolds in mathematics and physics" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 135 | DOI

[21] A Kirillov Jr, On G–equivariant modular categories, preprint (2004)

[22] J Maier, T Nikolaus, C Schweigert, Equivariant modular categories via Dijkgraaf–Witten theory, Adv. Theor. Math. Phys. 16 (2012) 289 | DOI

[23] M Müger, Galois extensions of braided tensor categories and braided crossed G–categories, J. Algebra 277 (2004) 256 | DOI

[24] L Müller, L Woike, Equivariant higher Hochschild homology and topological field theories, Homology Homotopy Appl. 22 (2020) 27 | DOI

[25] L Müller, L Woike, Parallel transport of higher flat gerbes as an extended homotopy quantum field theory, J. Homotopy Relat. Struct. 15 (2020) 113 | DOI

[26] E Riehl, Categorical homotopy theory, 24, Cambridge Univ. Press (2014) | DOI

[27] C Schweigert, L Woike, Extended homotopy quantum field theories and their orbifoldization, J. Pure Appl. Algebra 224 (2020) | DOI

[28] C H Taubes, Differential geometry: bundles, connections, metrics and curvature, 23, Oxford Univ. Press (2011) | DOI

[29] R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 | DOI

[30] V Turaev, Homotopy field theory in dimension 3 and crossed group-categories, preprint (2000)

[31] V Turaev, Homotopy quantum field theory, 10, Eur. Math. Soc. (2010) | DOI

[32] V Turaev, A Virelizier, On 3–dimensional homotopy quantum field theory, I, Internat. J. Math. 23 (2012) | DOI

[33] V Turaev, A Virelizier, On 3–dimensional homotopy quantum field theory, II : The surgery approach, Internat. J. Math. 25 (2014) | DOI

[34] D Yau, Colored operads, 170, Amer. Math. Soc. (2016)

[35] D Yau, Homotopical quantum field theory, preprint (2018)