What do homotopy algebras form?

TL;DR

This paper develops a homotopical framework for categories of $Cobar(C)$-algebras, using enriched categories, simplicial localization, and homotopy transfer techniques to model their $( rightarrow,1)$-categories.

Contribution

It constructs a simplicial model of the $( rightarrow,1)$-category of $Cobar(C)$-algebras, linking enriched categories, localization, and transfer theorems.

Findings

Constructed a symmetric monoidal category $LIE^{MC}$ for shifted L-infinity algebras.

Built a simplicial category $HoAlg^{ riangle}_C$ modeling the $( rightarrow,1)$-category of $Cobar(C)$-algebras.

Showed the Homotopy Transfer Theorem follows from the Goldman-Millson theorem.

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 $HoAlg^{\Delta}_C$ whose mapping spaces are Kan complexes. The simplicial category $HoAlg^{\Delta}_C$ gives us a particularly nice model of an $(\infty,1)$-category of $Cobar(C)$-algebras. We show that the homotopy category of $HoAlg^{\Delta}_C$ 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.