| Title:
|
On analytic contravariant functors on free groups (English) |
| Author:
|
Powell, Geoffrey |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
8 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
416-466 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Working over a field $k$ of characteristic zero, the category $F_{\omega}(gr^(op);k)$ of analytic contravariant functors on the category gr of finitely-generated free groups is shown to be equivalent to the category $F_{(Lie)}$ of representations of the $k$-linear category Cat Lie associated to the Lie operad. Two proofs are given of this result. The first uses the original Ginzburg-Kapranov approach to Koszul duality of binary quadratic operads and the fact that the category of analytic contravariant functors is Koszul. The second proof proceeds by making the equivalence explicit using the $k$-linear category $Cat USS^(u)$ associated to the operad $USS^(u)$ encoding unital associative algebras, which provides the ‘twisting bimodule’ between modules over Cat Lie and modules over $kgr^(op)$. A key ingredient is the Poincaré-Birkhoff-Witt theorem. Using the explicit formulation, it is shown how this equivalence reflects the tensor product on the category of analytic contravariant functors, relating this to the convolution product for representations of Cat Lie. (English) |
| Keyword:
|
Functor category |
| Keyword:
|
polynomial functor |
| Keyword:
|
free group |
| Keyword:
|
Lie operad |
| Keyword:
|
PROP |
| Keyword:
|
Poincaré-Birkhoff-Witt |
| MSC:
|
13D03 |
| MSC:
|
17B01 |
| MSC:
|
18A25 |
| MSC:
|
18M70 |
| MSC:
|
18M85 |
| idZBL:
|
Zbl 1560.18001 |
| idMR:
|
MR4835394 |
| DOI:
|
10.21136/HS.2024.15 |
| . |
| Date available:
|
2026-03-13T14:40:31Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153480 |
| . |
| Reference:
|
[1] Batanin, Michael, Markl, Martin: Operadic categories as a natural environment for Koszul duality.Compositionality, Vol. 5, Iss. 3, 46 MR 4599796 |
| Reference:
|
[2] Beilinson, A. A., Ginsburg, V. A., Schechtman, V. V.: Koszul duality.J. Geom. Phys., Vol. 5, Iss. 3, 317-350, https://doi.org/10.1016/0393-0440(88)90028-9 10.1016/0393-0440(88)90028-9 |
| Reference:
|
[3] Beilinson, Alexander, Ginzburg, Victor, Soergel, Wolfgang: Koszul duality patterns in representation theory.J. Amer. Math. Soc., Vol. 9, Iss. 2, 473-527, https://doi.org/10.1090/S0894-0347-96-00192-0, DOI:10.1090/S0894-0347-96-00192-0 10.1090/S0894-0347-96-00192-0 |
| Reference:
|
[4] Chmutov, S., Duzhin, S., Mostovoy, J.: Introduction to Vassiliev knot invariants.Cambridge University Press, Cambridge, DOI:10.1017/CBO9781139107846 MR 2962302, 10.1017/CBO9781139107846 |
| Reference:
|
[5] Djament, Aurélien, Pirashvili, Teimuraz, Vespa, Christine: Cohomologie des foncteurs polynomiaux sur les groupes libres.Doc. Math., Vol. 21, 205-222 MR 3505136 |
| Reference:
|
[6] Djament, Aurélien, Vespa, Christine: Sur l’homologie des groupes d’automorphismes des groupes libres à coefficients polynomiaux.Comment. Math. Helv., Vol. 90, Iss. 1, 33-58, http://dx.doi.org/10.4171/CMH/345, DOI:10.4171/CMH/345 MR 3317332, 10.4171/cmh/345 |
| Reference:
|
[7] Eilenberg, Samuel, Mac Lane, Saunders: On the groups H(Π,n). II. Methods of computation.Ann. of Math. (2), Vol. 60, 49-139, https://doi.org/10.2307/1969702, DOI:10.2307/1969702 10.2307/1969702 |
| Reference:
|
[8] Fresse, Benoit: Lie theory of formal groups over an operad.J. Algebra, Vol. 202, Iss. 2, 455-511, https://doi.org/10.1006/jabr.1997.7280, DOI:10.1006/jabr.1997.7280 10.1006/jabr.1997.7280 |
| Reference:
|
[9] Fresse, Benoit: Modules over operads and functors.Lecture notes in mathematics, Springer-Verlag, Berlin, https://doi.org/10.1007/978-3-540-89056-0, ISBN:978-3-540-89055-3, DOI:10.1007/978-3-540-89056-0 MR 2494775, 10.1007/978-3-540-89056-0 |
| Reference:
|
[10] Ginzburg, Victor, Kapranov, Mikhail: Koszul duality for operads.Duke Math. J., Vol. 76, Iss. 1, 203-272, https://doi.org/10.1215/S0012-7094-94-07608-4, DOI:10.1215/S0012-7094-94-07608-4 10.1215/S0012-7094-94-07608-4 |
| Reference:
|
[11] Hartl, Manfred, Pirashvili, Teimuraz, Vespa, Christine: Polynomial functors from algebras over a set-operad and nonlinear Mackey functors.Int. Math. Res. Not. IMRN, Iss. 6, 1461-1554, https://doi.org/10.1093/imrn/rnt242, DOI:10.1093/imrn/rnt242 MR 3340364, 10.1093/imrn/rnt242 |
| Reference:
|
[12] Kapranov, M., Manin, Yu.: Modules and Morita theorem for operads.Amer. J. Math., Vol. 123, Iss. 5, 811-838 Zbl 1001.18004, MR 1854112, 10.1353/ajm.2001.0033 |
| Reference:
|
[13] Kawazumi, Nariya, Vespa, Christine: On the wheeled PROP of stable cohomology of Aut(F_n) with bivariant coefficients.Algebr. Geom. Topol., Vol. 23, Iss. 7, 3089-3128, https://doi.org/10.2140/agt.2023.23.3089, DOI:10.2140/agt.2023.23.3089 MR 4647673, 10.2140/agt.2023.23.3089 |
| Reference:
|
[14] Loday, Jean-Louis, Vallette, Bruno: Algebraic operads.Grundlehren der mathematischen wissenschaften [fundamental principles of mathematical sciences], Springer, Heidelberg, https://doi.org/10.1007/978-3-642-30362-3, ISBN:978-3-642-30361-6, DOI:10.1007/978-3-642-30362-3 MR 2954392, 10.1007/978-3-642-30362-3 |
| Reference:
|
[15] Mac Lane, Saunders: Categorical algebra.Bull. Amer. Math. Soc., Vol. 71, 40-106, https://doi.org/10.1090/S0002-9904-1965-11234-4, DOI:10.1090/S0002-9904-1965-11234-4 10.1090/S0002-9904-1965-11234-4 |
| Reference:
|
[16] Macdonald, I. G.: Symmetric functions and Hall polynomials.Oxford classic texts in the physical sciences, The Clarendon Press, Oxford University Press, New York, ISBN:978-0-19-873912-8 MR 3443860 |
| Reference:
|
[17] Mitchell, Barry: Rings with several objects.Advances in Math., Vol. 8, 1-161, https://doi.org/10.1016/0001-8708(72)90002-3, DOI:10.1016/0001-8708(72)90002-3 10.1016/0001-8708(72)90002-3 |
| Reference:
|
[18] Pirashvili, Teimuraz: Hodge decomposition for higher order Hochschild homology.Ann. Sci. École Norm. Sup. (4), Vol. 33, Iss. 2, 151-179, http://dx.doi.org/10.1016/S0012-9593(00)00107-5, DOI:10.1016/S0012-9593(00)00107-5 10.1016/S0012-9593(00)00107-5 |
| Reference:
|
[19] Pirashvili, Teimuraz: On the PROP corresponding to bialgebras.Cah. Topol. Géom. Différ. Catég., Vol. 43, Iss. 3, 221-239 MR 1928233 |
| Reference:
|
[20] Powell, Geoffrey: On the Passi and the Mal’cev functors.arXiv:2309.07605, DOI:10.48550/arXiv.2309.07605 10.48550/arXiv.2309.07605 |
| Reference:
|
[21] Powell, Geoffrey: Outer functors and a general operadic framework.J. Algebra, Vol. 644, 526-562, https://doi.org/10.1016/j.jalgebra.2024.01.015, DOI:10.1016/j.jalgebra.2024.01.015 MR 4696223, 10.1016/j.jalgebra.2024.01.015 |
| Reference:
|
[22] Powell, Geoffrey, Vespa, Christine: Higher Hochschild homology and exponential functors.arXiv:1802.07574 MR 4929346 |
| Reference:
|
[23] Turchin, Victor, Willwacher, Thomas: Hochschild-Pirashvili homology on suspensions and representations of Out(F_n).Ann. Sci. Éc. Norm. Supér. (4), Vol. 52, Iss. 3, 761-795, https://doi.org/10.24033/asens.2396, DOI:10.24033/asens.2396 MR 3982870, 10.24033/asens.2396 |
| Reference:
|
[24] Vespa, Christine: Extensions between functors from free groups.Bull. Lond. Math. Soc., Vol. 50, Iss. 3, 401-419, https://doi.org/10.1112/blms.12091, DOI:10.1112/blms.12091 MR 3829729, 10.1112/blms.12091 |
| . |
