Some functorial factorizations for Quillen functors

TL;DR

The paper establishes new functorial factorizations for right Quillen functors, including lax monoidal cases, extending the theory of Quillen-Segal $ ext{O}$-algebras without requiring $ ext{M}$ to be combinatorial.

Contribution

It introduces non-trivial, functorial factorizations for right Quillen functors applicable to various model categories and extends the theory of Quillen-Segal $ ext{O}$-algebras.

Findings

Existence of functorial factorizations for right Quillen functors.

Extension of Quillen-Segal $ ext{O}$-algebra theory.

Applicability to lax monoidal right Quillen functors.

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. We also prove that these factorizations can be made for lax monoidal right Quillen functors. Given a monad, operad or a PROP(erad) $\mathcal{O}$, if we apply one of the factorizations to the forgetful functor $\mathcal{U} : \mathcal{O}-Alg(\mathcal{M}) \rightarrow \mathcal{M}$, we extend the theory of Quillen-Segal $\mathcal{O}$-algebras without the hypothesis of $\mathcal{M}$ being a combinatorial model category.