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relation between L-infinity algebras and dg-Lie algebras

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Higher algebra

Rational homotopy theory

Contents

Idea

The homotopy theory of L 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 L_\infty -algebras on those whose kk-ary brackets vanish for all k3k \geq 3,

(1)dgLieAlgebrasiL Algebras,AAAAL W(dgLieAlgebras)L W(i)L W(L Algebras) 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 WL_{\mathrm{W}} this inclusion is an equivalence of (∞,1)-categories.

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

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.).

References

Created on July 23, 2021 at 10:24:03. See the history of this page for a list of all contributions to it.