A Tale of Three Homotopies

Abstract

For a Koszul operad P, there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy P-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.2010 Mathematics Subject Classification. Primary 18G55; Secondary 18D50. Key words and phrases. homotopy algebras, homotopy morphisms, models for operads.

What do homotopy algebras form?

“…This work is motivated by Dmitry Tamarkin's answer [24] to Vladimir Drinfeld's question "What do dg categories form?" [10], and by papers [1], [8], [13], [15], and [18]. Here, we give an answer to the question "What do homotopy algebras form?…”

Higher algebra over the Leibniz operad

“…We proved [5] that these notions are all equivalent (we will briey come back to this fact later on). In the sequel, we prefer Quillen homotopies, i.e., for any xed V, W ∈ P ∞ -Alg, the homotopies or 2-morphisms are…”

“…In [5], we describe a fourth and a fth notion, cylinder homotopies and operadic homotopies, and prove the Theorem 6. The concepts of concordance, gauge homotopy, Quillen homotopy, cylinder homotopy, and operadic homotopy are equivalent.…”

“…In fact, the notion of operadic homotopy is homotopically equivalent to the others. To our knowledge, we give in [5] the rst explicit recipe to write a denition of operadic homotopy. This receipt is far from being simple.…”

“…This paper is based on talks given by the author at the conference`Glances@Manifolds Both, the present text and the underlying lectures, report on the joint works [5], [10], [11], [13] with Vladimir Dotsenko, Janusz Grabowski, Benoit Jubin, David Khudaverdian, Jian Qiu, and Kyosuke Uchino.…”