Quillen-Segal algebras and Stable homotopy theory

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