∞-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
V. Hinich, Homological algebra of homotopy algebras, Communications in algebra, 25(10) (1997), 3291-3323, pdf; erratum math.AG/0309453
Last revised on January 11, 2017 at 19:24:20. See the history of this page for a list of all contributions to it.