What do homotopy algebras form?

2014/06/06 by Dolgushev, Vasily A., Hoffnung, Alexander E., Rogers, Christopher L.
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1406.1751

Abstract

In paper arXiv:1406.1744, we constructed a symmetric monoidal category LIE^MC whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad C and show that algebras over the operad Cobar(C) naturally form a category enriched over LIE^MC. Following arXiv:1406.1744, we "integrate" this LIE^MC-enriched category to a simplicial category HoAlg^Δ_C whose mapping spaces are Kan complexes. The simplicial category HoAlg^Δ_C gives us a particularly nice model of an (∞,1)-category of Cobar(C)-algebras. We show that the homotopy category of HoAlg^Δ_C is the localization of the category of Cobar(C)-algebras and infinity morphisms with respect to infinity quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman-Millson theorem.

Related

• On an enhancement of the category of shifted L-infinity algebras
• Homotopy theory of homotopy algebras
• Homotopy Algebras for Operads
• Operads and Maurer-Cartan spaces
• Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces