Hugo Bacard | Searcharxiv

arXiv2020

Some functorial factorizations for Quillen functors

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.

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arXiv2018

Quillen-Segal algebras and Stable homotopy theory

Let $\mathscr{M}$ be a monoidal model category that is also combinatorial and left proper. If $\mathscr{O}$ is a monad, operad, properad, or a PROP; following Segal's ideas we develop a theory of Quillen-Segal $\mathscr{O}$-algebras and show that we have a Quillen equivalence between usual $\mathscr{O}$-algebras and Quillen-Segal algebras. We use this theory to get the stable homotopy category by a similar method as Hovey.

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