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

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.