Loop spaces in motivic homotopy theory
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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.
Decker, Marvin Glen (2006). Loop spaces in motivic homotopy theory. Doctoral dissertation, Texas A&M University. Available electronically from