A motivic homotopy theory without $\mathbb{A}^1$ invariance

TL;DR

This paper develops a new unstable motivic homotopy category with modulus that extends existing frameworks to include non-$\mathbb{A}^1$-invariant theories, broadening the scope of motivic homotopy theory.

Contribution

It constructs a motivic homotopy category with modulus for diagrams of schemes, enabling the study of non-$\mathbb{A}^1$-invariant cohomology theories.

Findings

Introduces a new unstable motivic homotopy category with modulus.

Extends the Morel-Voevodsky construction to diagrams of schemes.

Provides a framework for studying non-$\mathbb{A}^1$-invariant theories.

Abstract

In this paper, we continue the program initiated by Kahn-Saito-Yamazaki by constructing and studying an unstable motivic homotopy category with modulus, extending the Morel-Voevodsky construction from smooth schemes over a field $k$ to certain diagrams of schemes. We present this category as a candidate environment for studying representability problems for non $\mathbb{A}^1$-invariant generalized cohomology theories.