∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
symmetric monoidal (∞,1)-category of spectra
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
The homotopy theory of -algebras happens to be equivalent to that of dg-Lie algebras (over any ground field of characteristic zero). In fact, dg-Lie algebras, being chain complexes equipped with a binary bracket, manifestly form the full subcategory of the 1-category of -algebras on those whose -ary brackets vanish for all ,
and under simplicial localization this inclusion is an equivalence of (∞,1)-categories.
Conversely, this means that dg-Lie algebras are strict -algebras – namely those whose Jacobi identity is satisfied strictly, not just up to coherent higher homotopies – and that every -algebra may be rectified to a strict one.
When formulated in terms of operads this means that the canonical functor from the category of algebras over the Lie operad in chain complexes (which are the dg-Lie algebras) to that of algebras over any of its cofibrant resolutions (which are the -algebras) is, at least, a surjection on quasi-isomorphism classes. In this form the statement appears in Kriz and May 1995, p. 28.
This refines to an equivalence of homotopy theories as follows:
-algebras are the fibrant objects in the model structure on dg cocommutative coalgebras (Pridham 10, this Prop.);
dg-Lie algebras are fibrant objects in the model structure on dg-Lie algebras (trivially so)
and the inclusion (1) of the latter category of fibrant objects into the former extends to a right Quillen functor with left adjoint being rectification (this Prop.) and which constitutes a Quillen equivalence (Hinich 97, this Prop.).
Dan Quillen, Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725, pdf)
Igor Kriz, Peter May, p. 28 of: Operads, algebras, modules and motives, Astérisque 233, Société Mathématique de France (1995) (pdf, numdam:AST_1995__233__1_0)
Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)
Vladimir Hinich, Homological algebra of homotopy algebras, Communications in algebra, 25(10). 3291-3323 (1997)(arXiv:q-alg/9702015, Erratum: arXiv:math/0309453)
Created on July 23, 2021 at 14:24:03. See the history of this page for a list of all contributions to it.