Homotopy theory of simplicial presheaves in completely decomposable topologies

TL;DR

This paper introduces a new class of sites where a generalized Brown-Gersten homotopy theory for simplicial presheaves can be applied, bridging the gap between existing approaches.

Contribution

It defines a class of sites with completely decomposable topologies enabling a generalized Brown-Gersten homotopy theory for simplicial presheaves.

Findings

A new class of sites with completely decomposable topologies is introduced.

The generalized Brown-Gersten approach is applicable to these sites.

This framework extends homotopy theory to more general settings.

Abstract

There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a Noetherian space of finite dimension but does not extend to all sites. In this paper we define a class of sites for which a generalized version of the Brown-Gersten approach works.