What do homotopy algebras form?

Abstract

© 2015 Elsevier Inc. In paper [4], we constructed a symmetric monoidal category SLie∞MC whose objects are shifted (and filtered) L∞-algebras. Here, we fix a cooperad C and show that algebras over the operad Cobar(C) naturally form a category enriched over SLie∞MC. Following [4], we "integrate" this SLie∞MC-enriched category to a simplicial category HoAlgCδ whose mapping spaces are Kan complexes. The simplicial category HoAlgCδ gives us a particularly nice model of an (∞, 1)-category of Cobar(C)-algebras. We show that the homotopy category of HoAlgCδ is the localization of the category of Cobar(C)-algebras and ∞-morphisms with respect to ∞-quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman-Millson theorem.