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

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

#### Higher algebra

higher algebra

universal algebra

and

# Contents

## 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:

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 14:24:03. See the history of this page for a list of all contributions to it.