All known realizations of complete Lie algebras coincide

Félix, Yves¹ ; Fuentes, Mario² ; Murillo, Aniceto³

¹Département de Mathématiques, Université Catholique de Louvain, Louvain la-Neuve, Belgium
²CIMI, Insitut de Mathématiques, Université Paul Sabatier, Toulouse, France
³Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Málaga, Spain

Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 1155-1167

Résumé

We prove that for any reduced differential graded Lie algebra L, the classical Quillen geometrical realization 〈L〉Q is homotopy equivalent to the realization 〈L〉 = Hom ⁡ cdgl ⁡ (𝔏∙,L) constructed via the cosimplicial free complete differential graded Lie algebra 𝔏∙. As the latter is a deformation retract of the Deligne–Getzler–Hinich realization MC ⁡ ∙(L) we deduce that, up to homotopy, all known topological realization functors of complete differential graded Lie algebras coincide. Immediate consequences of our main result include an elementary proof of the Baues–Lemaire conjecture and the description of the Quillen realization as a representable functor.

Détail

DOI : 10.2140/agt.2025.25.1155

Keywords: CDGAs, Baues–Lemaire conjecture, Quillen realization

Bibliographie

[1] H J Baues, J M Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977) 219 | DOI

[2] A Berglund, Rational homotopy theory of mapping spaces via Lie theory for L∞-algebras, Homology Homotopy Appl. 17 (2015) 343 | DOI

[3] U Buijs, Y Félix, A Murillo, D Tanré, The infinity Quillen functor, Maurer–Cartan elements and DGL realizations, preprint (2017)

[4] U Buijs, Y Félix, A Murillo, D Tanré, Lie models in topology, 335, Birkhäuser (2020) | DOI

[5] U Buijs, Y Félix, A Murillo, D Tanré, Lie models of simplicial sets and representability of the Quillen functor, Israel J. Math. 238 (2020) 313 | DOI

[6] Y Félix, S Halperin, Enriched differential Lie algebras in topology, preprint (2022)

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

[8] E Getzler, Lie theory for nilpotent L∞-algebras, Ann. of Math. 170 (2009) 271 | DOI

[9] V Hinich, Descent of Deligne groupoids, Int. Math. Res. Not. 1997 (1997) 223 | DOI

[10] M Majewski, Rational homotopical models and uniqueness, 682, Amer. Math. Soc. (2000) | DOI

[11] J P May, Simplicial objects in algebraic topology, 11, Van Nostrand (1967)

[12] J P Pridham, Pro-algebraic homotopy types, Proc. Lond. Math. Soc. 97 (2008) 273 | DOI

[13] D Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205 | DOI

[14] D Robert-Nicoud, Representing the deformation ∞-groupoid, Algebr. Geom. Topol. 19 (2019) 1453 | DOI

[15] D Robert-Nicoud, B Vallette, Higher Lie theory, preprint (2020)

[16] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 269 | DOI