Operad

On Weak Lie 3-Algebras

Strongly homotopy Lie algebras or $\mathrm{L}_\infty$-algebras are a generalization of (dg) Lie algebras with better homotopy properties. In particular, a version of the homotopy transfer theorem holds. This is, however, only true when working over a field of characteristic $0.$ When working over an arbitrary unital commutative ring, one must generalize the definition even further, relaxing the skew-symmetry of the brackets up to homotopy in addition to the Jacobi identity.

I just finished my article “On weak Lie $3$-algebras,” in which I consider precisely this problem. I describe a general step by step approach and give explicit definitions for the case of underlying $3$-term complexes. In this post, I’ll try to give a rough idea of the approach used and a quick overview of the main results.

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.