Loop spaces in motivic homotopy theory

Abstract

In topology loop spaces can be understood combinatorially using algebraic theories. This approach can be extended to work for certain model structures on categories of presheaves over a site with functorial unit interval objects, such as topological spaces and simplicial sheaves of smooth schemes at finite type. For such model categories a new category of algebraic theories with a proper cellular simplicial model structure can be defined. This model structure can be localized in a way compatible with left Bousfield localizations of the underlying category of presheaves to yield a Motivic model structure for algebraic theories. As in the topological context, the model structure is Quillen equivalent to a category of loop spaces in the underlying category.