∞-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
Given an ambient additive symmetric monoidal category, the Lie operad is the operad whose algebras over an operad are Lie algebras in that category. It is a quadratic operad whose Koszul dual is the operad for commutative algebras.
(This Koszul duality is what makes L-∞ algebras be equivalent to (semifree) differential graded coalgebras.)
In the case of the monoidal category of chain complexes, we also say dg-Lie-operad. Its cofibrant resolution in the model structure on operads is the L-infinity operad whose algebras over an operad are L-∞-algebras.
Lie operad
Vladimir Hinich: Homological algebra of homotopy algebras, Communications in algebra 25 10 (1997), 3291–3323 [pdf; erratum math.AG/0309453]
In spectra:
Last revised on June 17, 2026 at 04:40:11. See the history of this page for a list of all contributions to it.