Homotopy Theory of Homotopy Algebras

The goal of this paper is to study the homotopy theory of homotopy algebras over a Koszul operad with their infinity morphisms. The method consists in endowing the category of coalgebras over the Koszul dual cooperad with a model category structure. CONTENTS Introduction 1 1. Recollections 2 2. Model category structure for coalgebras 5 3. Conclusion 14 Appendix A. A technical lemma 15 References 16 INTRODUCTION In his seminal paper [Qui69], Quillen developed Rational Homotopy Theory by proving that several homotopy categories are equivalent. These homotopy categories are defined as localization with respect to weak equivalences, which are often quasi-isomorphisms. However, this result relies on a strong connectivity assumption. Hinich [Hin01] and then Lefevre-Hasegawa [LH03] showed how to bypass this assumption by considering on cocommutative and coassociative coalgebras respectively a class of weak equivalences, which is stricly included inside the class of quasi-isomorphisms. In the present paper, we extend Hinich and Lefevre-Hasegawa results to any category of conilpotent coalgebras over the Koszul dual of a Koszul operad: we endow them with a model category structure. This method allows us to prove the following main result, which gives a presentation of the homotopy categories of differential graded algebras over a Koszul operad in terms of∞-morphisms. Theorem (Theorem 3.1 and 3.2). Let P be a Koszul operad. Any∞-quasi-isomorphism of P∞-algebras admits a homotopy inverse. The following categories are equivalent Ho(dg P-alg) ' Ho(∞-P∞-alg) ' ∞-P∞-alg/ ∼h ' ∞-P-alg/ ∼h . Layout. The paper is organized as follows. We begin by some recollections on operadic homological algebra and on the model category for algebras. In the second section, we endow the category of coalgebras over a Koszul dual cooperad with a model category structure. Prerequisites. The reader is supposed to be familiar with the notion of an operad and operadic homological algebra, for which we refer to the book [LV12]. In the present paper, we use the same notations as used in this reference. Framework. Throughout this paper, we work over a field K of characteristic 0. Every chain complex is Z-graded with homological degree convention, i.e. with degree −1 differential. All the S-modules M = {M(n)}n∈N are reduced, that is M(0) = 0.