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.

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.

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.