A motivic homotopy theory without A1-invariance

In this paper, we continue the program initiated by Kahn\u2013Saito\u2013Yamazaki by constructing and studying an unstable motivic homotopy category with modulus MH(k), extending the Morel\u2013Voevodsky 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 A1-invariant generalized cohomology theories