What do homotopy algebras form?

In paper [4], we constructed a symmetric monoidal category SLie∞MC whose objects are shifted (and filtered) L∞L∞-algebras. Here, we fix a cooperad CC and show that algebras over the operad Cobar(C)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)(∞,1)-category of Cobar(C)Cobar(C)-algebras. We show that the homotopy category of HoAlgCΔ is the localization of the category of Cobar(C)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.