Homotopy theory of presheaves of Gamma-spaces

TL;DR

This paper develops a framework for stable homotopy theory of presheaves of Gamma-spaces, enabling algebraic structures to be studied within a homotopical context.

Contribution

It constructs stable model structures on presheaves of Gamma-spaces parametrized by local model structures, extending homotopical methods to algebraic objects.

Findings

Stable model structures on presheaves of Gamma-spaces are constructed.

Monoidal local model structures induce monoidal stable model structures.

Lifting of model structures to categories of algebras and modules is achieved.

Abstract

We consider the category of presheaves of Gamma-spaces, or equivalently, of Gamma-objects in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, the corresponding stable model structure on presheaves of Gamma-spaces is monoidal and satisfies the monoid axiom. This allows us to lift the stable model structures to categories of algebras and modules over commutative algebras.