Hochschild and cotangent complexes of operadic algebras
HTML articles powered by AMS MathViewer
- by Truong Hoang;
- Trans. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/tran/9447
- Published electronically: April 25, 2025
- HTML | PDF | Request permission
Abstract:
We make use of the cotangent complex formalism developed by Lurie to formulate Quillen cohomology of algebras over an enriched operad. Additionally, we introduce a spectral Hochschild cohomology theory for enriched operads and algebras over them. We prove that both the Quillen and Hochschild cohomologies of algebras over an operad can be controlled by the corresponding cohomologies of the operad itself. When passing to the category of simplicial sets, we assert that both these cohomology theories for operads, as well as their associated algebras, can be calculated in the same framework of spectrum valued functors on the twisted arrow $\infty$-category of the operad of interest. Moreover, we provide a convenient cofiber sequence relating the Hochschild and cotangent complexes of an $E_n$-space, establishing an unstable analogue of a significant result obtained by Francis and Lurie.References
- Gregory Arone and Victor Turchin, On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol. 18 (2014), no. 3, 1261–1322. MR 3228453, DOI 10.2140/gt.2014.18.1261
- Michael Batanin and David White, Left Bousfield localization without left properness, J. Pure Appl. Algebra 228 (2024), no. 6, Paper No. 107570, 23. MR 4670515, DOI 10.1016/j.jpaa.2023.107570
- Hans Joachim Baues and Günther Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187–211. MR 814176, DOI 10.1016/0022-4049(85)90008-8
- Clemens Berger and Ieke Moerdijk, On the derived category of an algebra over an operad, Georgian Math. J. 16 (2009), no. 1, 13–28. MR 2527612
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 365573
- G. Caviglia, A model structure for enriched coloured operads, Preprint, arXiv:1401.6983, 2014.
- Julien Ducoulombier, Benoit Fresse, and Victor Turchin, Projective and Reedy model category structures for (infinitesimal) bimodules over an operad, Appl. Categ. Structures 30 (2022), no. 5, 825–920. MR 4473902, DOI 10.1007/s10485-022-09675-z
- John Francis, The tangent complex and Hochschild cohomology of $\scr E_n$-rings, Compos. Math. 149 (2013), no. 3, 430–480. MR 3040746, DOI 10.1112/S0010437X12000140
- Benoit Fresse, Modules over operads and functors, Lecture Notes in Mathematics, vol. 1967, Springer-Verlag, Berlin, 2009. MR 2494775, DOI 10.1007/978-3-540-89056-0
- Javier J. Gutiérrez and Rainer M. Vogt, A model structure for coloured operads in symmetric spectra, Math. Z. 270 (2012), no. 1-2, 223–239. MR 2875831, DOI 10.1007/s00209-010-0794-2
- Yonatan Harpaz, Joost Nuiten, and Matan Prasma, The abstract cotangent complex and Quillen cohomology of enriched categories, J. Topol. 11 (2018), no. 3, 752–798. MR 3989430, DOI 10.1112/topo.12074
- Yonatan Harpaz, Joost Nuiten, and Matan Prasma, Tangent categories of algebras over operads, Israel J. Math. 234 (2019), no. 2, 691–742. MR 4040842, DOI 10.1007/s11856-019-1933-z
- Yonatan Harpaz, Joost Nuiten, and Matan Prasma, The tangent bundle of a model category, Theory Appl. Categ. 34 (2019), Paper No. 33, 1039–1072. MR 4020832
- Truong Hoang, Quillen cohomology of enriched operads, Adv. Math. 465 (2025), Paper No. 110151, 91. MR 4864698, DOI 10.1016/j.aim.2025.110151
- T. Hoang, Operadic Dold-Kan correspondence and mapping spaces between enriched operads, Preprint, arXiv:2312.07906, 2023.
- G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58–67. MR 11076, DOI 10.2307/1969145
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- Jean-Louis Loday, Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by María O. Ronco; Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR 1600246, DOI 10.1007/978-3-662-11389-9
- Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392, DOI 10.1007/978-3-642-30362-3
- J. Lurie, Higher algebra, Preprint, 2011, http://www.math.ias.edu/~lurie/.
- Sergei Merkulov and Bruno Vallette, Deformation theory of representations of prop(erad)s. II, J. Reine Angew. Math. 636 (2009), 123–174. MR 2572248, DOI 10.1515/CRELLE.2009.084
- Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 151–179 (English, with English and French summaries). MR 1755114, DOI 10.1016/S0012-9593(00)00107-5
- T. Pirashvili and B. Richter, Hochschild and cyclic homology via functor homology, $K$-Theory 25 (2002), no. 1, 39–49. MR 1899698, DOI 10.1023/A:1015064621329
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 223432
- Charles W. Rezk, Spaces of algebra structures and cohomology of operads, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2716655
- Alan Robinson, Gamma homology, Lie representations and $E_\infty$ multiplications, Invent. Math. 152 (2003), no. 2, 331–348. MR 1974890, DOI 10.1007/s00222-002-0272-5
- Alan Robinson, $E_\infty$ obstruction theory, Homology Homotopy Appl. 20 (2018), no. 1, 155–184. MR 3775354, DOI 10.4310/HHA.2018.v20.n1.a10
- Stefan Schwede and Brooke Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003), 287–334. MR 1997322, DOI 10.2140/agt.2003.3.287
- M. Spitzweck, Operads, algebras and modules in general model categories, Preprint, arXiv:math/0101102, 2001.
- Bertrand Toën, The homotopy theory of $dg$-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615–667. MR 2276263, DOI 10.1007/s00222-006-0025-y
Bibliographic Information
- Truong Hoang
- Affiliation: Department of Mathematics, Hanoi FPT University, Vietnam
- ORCID: 0009-0001-7976-5385
- Email: truonghm@fe.edu.vn
- Received by editor(s): January 11, 2024
- Received by editor(s) in revised form: December 22, 2024, and February 17, 2025
- Published electronically: April 25, 2025
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
- MSC (2020): Primary 55P42, 18M60, 18N60, 18M75
- DOI: https://doi.org/10.1090/tran/9447