nLab L-infinity operad

Idea

The $L_\infty$-operad is a cofibrant resolution of the Lie operad in the model structure on operads in the category of chain complexes.

Its algebras over an operad are L-infinity algebras.

Related concepts

• A-∞ operad

• E-∞ operad

• model structure for L-∞ algebras

References

In chain complexes

Discussion as an operad in chain complexes:

Original articles include

• Takashi Kimura, Jim Stasheff, Alexander Voronov, around Def. 2.1 of: On operad structures of moduli spaces and string theory, Comm. Math. Phys. Volume 171, Number 1 (1995), 1-25 (euclid:1104273401)

Review includes

• Jeroen Maes, around Def. 5.6.4 of Derived homotopy algebras, PhD thesis 2016 (web)

See also

• Igor Kriz, Peter May, around p. 28 of Operads, algebras, modules and motives (pdf)

In spectra

Discussion as an operad in spectra (in stable homotopy theory) is in

• Gijs Heuts, around def. 4.12 of Lie algebras and $v_n$-periodic spaces (arXiv:1803.06325)

based on

• Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. , 9:833–933, 2005 (arXiv:math/0501429)

• Michael Ching, Bar-cobar duality for operads in stable homotopy theory, Journal of Topology, 2012 (arXiv:1009.5034)