We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad ${\mathcal{O}}$ to associative algebras in the category of homotopy ${\mathcal{O}}$-algebras. As an example, we identify the category of $\mathbb{P}_{n+1}$-algebras with the category of associative algebras in $\mathbb{P}_{n}$-algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.

We realize the multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.