Enriched Simplicial Presheaves and the Motivic Homotopy Category

TL;DR

This paper develops models for the motivic homotopy category using simplicial presheaves that are homotopy invariant, eliminating the need to invert the affine line for such models.

Contribution

It introduces a new approach to modeling the motivic homotopy category via enriched simplicial presheaves that are homotopy invariant.

Findings

Constructed models based on simplicial functors from smooth schemes.

Models are homotopy invariant without inverting the affine line.

Provides a new framework for motivic homotopy theory.

Abstract

We construct models for the motivic homotopy category based on simplicial functors from smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and therefore one does not have to invert the affine line in order to get a model for the motivic homotopy category.