Quillen-Segal objects and structures: an overview

TL;DR

This paper develops a homotopy theory for Quillen-Segal objects and structures in model categories, generalizing classical algebraic structures with weak equivalences and comparing them to traditional algebraic frameworks.

Contribution

It introduces the concept of QS-algebras and structures within model categories, extending classical algebraic notions to a homotopical context with a new theoretical framework.

Findings

Constructs a homotopy theory for QS-objects and structures.

Shows the relation between QS-structures and classical algebraic structures.

Utilizes Smith's theorem for localization in model categories.

Abstract

Let $\mathscr{M}$ be a combinatorial and left proper model category, possibly with a monoidal structure. If $\mathscr{O}$ is either a monad on $\mathscr{M}$ or an operad enriched over $\mathscr{M}$, define a QS-algebra in $\mathscr{M}$ to be a weak equivalence $\mathscr{F}: s(\mathscr{F}) \xrightarrow{\sim}t(\mathscr{F})$ such that the target $t(\mathscr{F})$ is an $\mathscr{O}$-algebra in the usual sense. A classical $\mathscr{O}$-algebra is a QS-algebra supported by an isomorphism $\mathscr{F}$. A QS-structure $\mathscr{F}$ is also a weak equivalence such that $t(\mathscr{F})$ has a structure, e.g, Hodge, twistorial, schematic, sheaf, etc. We build a homotopy theory of these objects and compare it with that of usual $\mathscr{O}$-algebras/structures. Our results rely on Smith's theorem on left Bousfield localization for combinatorial and left proper model categories. These ideas are derived from the theory of co-Segal algebras and categories.