Coextension of scalars in operad theory (Q2126066)

A morphism \(f : \mathcal{P} \to \mathcal{Q}\) of operads induces a functor \(f^* : \mathcal{Q}-alg \to \mathcal{P}-alg\) in the opposite direction between the respective categories of algebras. Under mild conditions in the underlying symmetric monoidal category, \(f^*\) admits a left adjoint. An interesting question is when the functor \(f^*\) admits a right adjoint. The main result of this paper (Theorems 3.4 and 3.8) answers this question completely by providing a necessary and sufficient condition that guarantees that \(f^*\) admits a right adjoint. This condition says roughly that each operation in \(\mathcal{Q}\) can be written in an essentially unique way as a unary operation followed by an opertion in \(\mathcal{P}\).