| Title:
|
Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules (English) |
| Author:
|
Campos, Ricardo |
| Author:
|
Ducoulombier, Julien |
| Author:
|
Idrissi, Najib |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
5 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
310-383 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)-operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman–Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements. (English) |
| Keyword:
|
operads |
| Keyword:
|
bimodules over operads |
| Keyword:
|
bar construction |
| Keyword:
|
model category |
| Keyword:
|
homotopy theory |
| Keyword:
|
cofibrant and fibrant resolutions |
| MSC:
|
18M70 |
| MSC:
|
18M75 |
| MSC:
|
18N40 |
| idZBL:
|
Zbl 1483.18024 |
| idMR:
|
MR4367224 |
| DOI:
|
10.21136/HS.2021.09 |
| . |
| Date available:
|
2026-03-13T05:38:30Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153441 |
| . |
| Reference:
|
[1] Arone, Greg, Ching, Michael: Operads and chain rules for the calculus of functors.Astérisque 338, ISBN:978-2-85629-308-9 MR 2840569 |
| Reference:
|
[2] Arone, Gregory, Turchin, Victor: On the rational homology of high dimensional analogues of spaces of long knots.Geom. Topol., Vol. 18, Iss. 3, 1261-1322, https://arxiv.org/abs/1105.1576, DOI:10.2140/gt.2014.18.1261 MR 3228453, 10.2140/gt.2014.18.1261 |
| Reference:
|
[3] Batanin, Michael, Markl, Martin: Operadic categories and duoidal Deligne’s conjecture.Adv. Math., Vol. 285, 1630-1687, https://arxiv.org/abs/1404.3886, DOI:10.1016/j.aim.2015.07.008 MR 3406537, 10.1016/j.aim.2015.07.008 |
| Reference:
|
[4] Berger, Clemens, Moerdijk, Ieke: Axiomatic homotopy theory for operads.Comment. Math. Helv., Vol. 78, Iss. 4, 805-831, DOI:10.1007/s00014-003-0772-y MR 2016697, 10.1007/s00014-003-0772-y |
| Reference:
|
[5] Berger, Clemens, Moerdijk, Ieke: The Boardman–Vogt resolution of operads in monoidal model categories.Topology, Vol. 45, Iss. 5, 807-849, DOI:10.1016/j.top.2006.05.001 MR 2248514, 10.1016/j.top.2006.05.001 |
| Reference:
|
[6] Boardman, J. Michael, Vogt, Rainer M.: Homotopy invariant algebraic structures on topological spaces.Lecture notes in mathematics 347, Springer, ISBN:978-3-540-06479-4, DOI:10.1007/BFb0068547 10.1007/BFb0068547 |
| Reference:
|
[7] Boardman, J. Michael, Vogt, Rainer M.: Homotopy-everything H-spaces.Bull. Amer. Math. Soc., Vol. 74, 1117-1122, DOI:10.1090/S0002-9904-1968-12070-1 10.1090/S0002-9904-1968-12070-1 |
| Reference:
|
[8] Brito, Pedro, Weiss, Michael S.: Spaces of smooth embeddings and configuration categories.J. Topol., Vol. 11, Iss. 1, 65-143, https://arxiv.org/abs/1502.01640, DOI:10.1112/topo.12048 MR 3784227, 10.1112/topo.12048 |
| Reference:
|
[9] Ching, Michael: Bar constructions for topological operads and the Goodwillie derivatives of the identity.Geom. Topol., Vol. 9, 833-933 (electronic), https://arxiv.org/abs/math/0501429, DOI:10.2140/gt.2005.9.833 MR 2140994, 10.2140/gt.2005.9.833 |
| Reference:
|
[10] Ching, Michael: Bar-cobar duality for operads in stable homotopy theory.J. Topol., Vol. 5, Iss. 1, 39-80, https://doi.org/10.1112/jtopol/jtr027, DOI:10.1112/jtopol/jtr027 MR 2897049, 10.1112/jtopol/jtr027 |
| Reference:
|
[11] Ducoulombier, Julien: Delooping of high-dimensional spaces of string links.https://arxiv.org/abs/1809.00682, pre-published |
| Reference:
|
[12] Ducoulombier, Julien: From maps between coloured operads to Swiss-Cheese algebras.Ann. Inst. Fourier, Vol. 68, Iss. 2, 661-724, https://arxiv.org/abs/1603.07162, DOI:10.5802/aif.3175 MR 3803116, 10.5802/aif.3175 |
| Reference:
|
[13] Ducoulombier, Julien: From maps between coloured operads to Swiss-Cheese algebras.Ann. Inst. Fourier, Vol. 68, Iss. 2, 661-724, http://aif.cedram.org/item?id=AIF_2018__68_2_661_0 MR 3803116, 10.5802/aif.3175 |
| Reference:
|
[14] Ducoulombier, Julien, Fresse, Benoit, Turchin, Victor: Projective and reedy model category structures for (infinitesimal) bimodules over an operad.https://arxiv.org/abs/1911.03890, pre-published MR 4473902 |
| Reference:
|
[15] Ducoulombier, Julien, Turchin, Victor: Delooping the functor calculus tower.https://arxiv.org/abs/1708.02203, pre-published MR 4442678 |
| Reference:
|
[16] Elmendorf, A. D., Kriz, I., Mandell, M. A., May, J. P.: Rings, modules, and algebras in stable homotopy theory.Mathematical surveys and monographs, American Mathematical Society, ISBN:0-8218-0638-6 |
| Reference:
|
[17] Fresse, Benoit: Koszul duality of operads and homology of partition posets.Homotopy theory: Relations with algebraic geometry, group cohomology, and algebraic K-theory, pages 115-215, Contemp. math. 346 MR 2066499 |
| Reference:
|
[18] Fresse, Benoit: Homotopy of operads and Grothendieck–Teichmüller groups.Mathematical surveys and monographs 217, Americal Mathematical Society, ISBN:978-1-4704-3481-6 MR 3643404 |
| Reference:
|
[19] Fresse, Benoit: Homotopy of operads and Grothendieck–Teichmüller groups.Mathematical surveys and monographs 217, Americal Mathematical Society, ISBN:978-1-4704-3482-3 MR 3616816 |
| Reference:
|
[20] Fresse, Benoit, Turchin, Victor, Willwacher, Thomas: The rational homotopy of mapping spaces of E_n operads.https://arxiv.org/abs/1703.06123, pre-published MR 3870769 |
| Reference:
|
[21] Fresse, Benoit, Willwacher, Thomas: Rational homotopy theory of operad modules.in preparation |
| Reference:
|
[22] Goodwillie, Thomas G., Weiss, Michael: Embeddings from the point of view of immersion theory : Part II.Geom. Topol., Vol. 3, 103-118 (electronic), DOI:10.2140/gt.1999.3.103 10.2140/gt.1999.3.103 |
| Reference:
|
[23] Livernet, Muriel: Koszul duality of the category of trees and bar constructions for operads.Operads and universal algebra, pages 107-138, Nankai ser. Pure appl. Math. Theoret. phys. 9 MR 3013085 |
| Reference:
|
[24] Loday, Jean-Louis, Vallette, Bruno: Algebraic operads.Grundlehren der mathematischen wissenschaften 346, Springer, ISBN:978-3-642-30361-6, DOI:10.1007/978-3-642-30362-3 MR 2954392, 10.1007/978-3-642-30362-3 |
| Reference:
|
[25] Markl, Martin, Shnider, Steve, Stasheff, Jim: Operads in algebra, topology and physics.Mathematical surveys and monographs 96, Amer. Math. Soc., ISBN:0-8218-2134-2 MR 1898414 |
| Reference:
|
[26] May, J. Peter: The geometry of iterated loop spaces.Lectures notes in mathematics 271, Springer-Verlag, DOI:10.1007/BFb0067491 10.1007/BFb0067491 |
| Reference:
|
[27] Salvatore, Paolo: Configuration operads, minimal models and rational curves..PhD thesis, University of Oxford |
| . |
