On $B_{\infty}$-algebras

$B_{\infty}$-algebras --- anecdotally, Baues-infinity algebras --- combine the structure of $A_{\infty}$-algebras and multibrace algebras, corresponding to endowing the bar construction with compatible differentials and products respectively. One important example of $B_\infty$ structures was given by Baues in his work on iterated loop spaces: the bar construction on the cochains on a simplicial set has a canonical multiplication, dual to comultiplication given by the (Serre) diagonal approximation on the (cubical) cobar construction.  Another example is the $B_\infty$-algebra structure on the Hochschild cochain complex, and the relation of $B_\infty$- and $G_\infty$ algebras was central in the resolutions of Deligne's Hochschild cohomology conjecture.

Loday and Ronco, in their work generalising the Milnor-Moore theorem to non-cocommutative Hopf algebras, introduced a universal enveloping functor from multibrace algebras to 2-associative algebras and showed that the space of primitives Prim$(H)$ of a cofree Hopf algebra has a multibrace structure whose universal enveloping recovers $H$.  In this talk we present work in progress (with I Gálvez and M Ronco) to extend these ideas to $B_{\infty}$-algebras.