Symmetric Homotopy Theory for Operads

My first article (joint work with Bruno Vallette) has just appeared on arXiv! It deals with various constructions related to the homotopy theory of dg operads over arbitrary unital commutative rings: in particular, we define a higher cobar-bar adjunction and show that it provides us with a new cofibrant replacement functor, and introduce a notion of higher homotopy operads. Here, I would like to take the time to explain the motivation behind this work, sketch the approach we used, and talk about the meaning of the main results.

Algebraic Structures up to Homotopy

Roughly, by an algebraic structure I mean a space equipped with a bunch of structure maps which are subject to certain relations. In general, algebraic structures are rigid and do not behave well with respect to homotopy operations on their underlying space. However, some algebraic structures are sufficiently flexible and much better behaved in this regard. We call these algebraic structure up to homotopy. Sounds vague and confusing? In this post, we’ll consider a concrete and easy example: associative algebras.