nLab relation between L-infinity algebras and dg-Lie algebras

Idea

The homotopy theory of $L_\infty$-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 $L_\infty$-algebras on those whose $k$-ary brackets vanish for all $k \geq 3$,

(1)$$dgLieAlgebras \xhookrightarrow{\;\;i\;\;} L_\infty Algebras \,, {\phantom{AAAA}} L_{\mathrm{W}} \big( dgLieAlgebras \big) \underoverset {\simeq} {\;\;L_{\mathrm{W}}(i)\;\;} {\longrightarrow} L_{\mathrm{W}} \big( L_\infty Algebras \big) \,$$

and under simplicial localization $L_{\mathrm{W}}$ this inclusion is an equivalence of (∞,1)-categories.

Conversely, this means that dg-Lie algebras are strict $L_\infty$-algebras – namely those whose Jacobi identity is satisfied strictly, not just up to coherent higher homotopies – and that every $L_\infty$-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 $L_\infty$-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:

• $L_\infty$-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.).

Related entries

• model structure for $L_\infty$-algebras

• rectification

• rational homotopy theory

References

• 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)