What do homotopy algebras form?

Abstract

In paper arXiv:1406.1744, we constructed a symmetric monoidal category $LI{E}^{MC}$ 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 $LI{E}^{MC}$. Following arXiv:1406.1744, we "integrate" this $LI{E}^{MC}$-enriched category to a simplicial category $HoAl{g}_C^{\Delta }$ whose mapping spaces are Kan complexes. The simplicial category $HoAl{g}_C^{\Delta }$ gives us a particularly nice model of an $(\infty ,1)$-category of $Cobar(C)$-algebras. We show that the homotopy category of $HoAl{g}_C^{\Delta }$ 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.

The final version will appear in Advances in Mathematics. Comments are still welcome