Swiss-Cheese operad and Drinfeld center. (English) Zbl 1422.55021

Summary: We build a model in groupoids for the Swiss-Cheese operad, based on parenthesized permutations and braids. We relate algebras over this model to the classical description of algebras over the homology of the Swiss-Cheese operad. We extend our model to a rational model for the Swiss-Cheese operad, and we compare it to the model that we would get if the Swiss-Cheese operad were formal.


55P48 Loop space machines and operads in algebraic topology
18D50 Operads (MSC2010)
Full Text: DOI arXiv


[1] Ayala, D.; Francis, J.; Tanaka, H. L., Factorization homology of stratified spaces, Selecta Math. (N.S.), 23, 293-362, (2017) · Zbl 1365.57037
[2] J. M. Boardman and R. M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics, no. 347, Springer, Berlin, 1973. · Zbl 0285.55012
[3] F. R. Cohen, The homology of Cn+1 spaces, n = 0, in The homology of iterated loop spaces, Lecture Notes in Mathematics, no. 533, Springer, 1976, pp. 207-351.
[4] Drinfel’d, V. G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with gal(Q/Q), Algebra i Analiz, 2, 149-181, (1990) · Zbl 0718.16034
[5] Epstein, D. B. A., Functors between tensored categories, Invent. Math., 1, 221-228, (1966) · Zbl 0146.02502
[6] B. Fresse, Modules over operads and functors, Lecture Notes in Mathematics, no. 1967, Springer-Verlag, 2009. · Zbl 1178.18007
[7] B. Fresse, Homotopy of Operads and Grothendieck-Teichmüller Groups, Mathematical Surveys and Monographs, Vol. 217, American Mathematical Society, Providence, RI, in press. · Zbl 1375.55007
[8] B. Fresse and T. Willwacher, The intrinsic formality of En-operads, 2015, Preprint.
[9] Ginot, G.; Calaque, D. (ed.); Strobl, T. (ed.), Notes on factorization algebras, factorization homology and applications, 429-552, (2015), Springer International Publishing · Zbl 1315.81092
[10] Hoefel, E., OCHA and the swiss-cheese operad, J. Homotopy Relat. Struct., 4, 123-151, (2009) · Zbl 1185.18010
[11] Hoefel, E.; Livernet, M., Open-closed homotopy algebras and strong homotopy Leibniz pairs through Koszul operad theory, Lett. Math. Phys., 101, 195-222, (2012) · Zbl 1276.18017
[12] Joyal, A.; Street, R., Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra, 71, 43-51, (1991) · Zbl 0726.18004
[13] Kajiura, H.; Stasheff, J., Homotopy algebras inspired by classical open-closed string field theory, Comm. Math. Phys., 263, 553-581, (2006) · Zbl 1125.18012
[14] Kontsevich, M., Operads and motives in deformation quantization, Lett. Math. Phys., 48, 35-72, (1999) · Zbl 0945.18008
[15] Livernet, M., Non-formality of the swiss-cheese operad, J. Topol., 8, 1156-1166, (2015) · Zbl 1333.55009
[16] S. Mac Lane, Categories for the Working Mathematician, 2 ed., Graduate Texts in Mathematics, no. 5, Springer-Verlag, 1998. · Zbl 0906.18001
[17] Majid, S., Representations, duals and quantum doubles of monoidal categories, 197-206, (1991) · Zbl 0762.18005
[18] J. P. May, The Geometry of Iterated Loop Spaces, Lectures Notes in Mathematics, no. 271, Springer-Verlag, 1972. · Zbl 0244.55009
[19] Severa, P.; Willwacher, T., Equivalence of formalities of the little discs operad, Duke Math. J., 160, 175-206, (2011) · Zbl 1241.18008
[20] Tamarkin, D. E., Formality of chain operad of little discs, Lett. Math. Phys., 66, 65-72, (2003) · Zbl 1048.18007
[21] Voronov, A. A., The swiss-cheese operad, 365-373, (1999) · Zbl 0946.55005
[22] T. Willwacher, Models for the n-Swiss Cheese operads, 2015, Preprint.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.