Functorial Factorizations in the Category of Model Categories

Abstract

We prove that any right Quillen functor between arbitrary model categories admits non trivial functorial factorizations that are similar to those of a model structure. Given a monad, operad or a PROP(erad) $$\mathcal {O}$$ , if we apply one of the factorizations to the forgetful functor $$\textsf {U}: \mathcal {O}{\text {-Alg}}(\textsf {M}) \longrightarrow \textsf {M}$$ , we extend the theory of Quillen–Segal $$\mathcal {O}$$ -algebras initiated in Bacard (Higher Struct 4(1):57–114, 2020), without the hypothesis of $$\textsf {M}$$ being a combinatorial model category.