  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.
