Quillen-Segal algebras and Stable homotopy theory

TL;DR

This paper develops a new framework for Quillen-Segal algebras associated with various algebraic structures in monoidal model categories, establishing a Quillen equivalence and deriving the stable homotopy category using this approach.

Contribution

It introduces a theory of Quillen-Segal $ extit{O}$-algebras for monads, operads, properads, and PROPs, and proves a Quillen equivalence with traditional $ extit{O}$-algebras.

Findings

Established a Quillen equivalence between usual and Quillen-Segal $ extit{O}$-algebras.

Derived the stable homotopy category via this new algebraic framework.

Extended Segal's ideas to a broad class of algebraic structures in homotopy theory.

Abstract

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.