Cogroups in algebras over an operad are free algebras

Abstract

Let $\mathcal{P}$ be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete $\mathcal{P}$-algebras. In this article, we show that R is necessarily the completion of a free $\mathcal{P}$-algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad.