Title: A motivic homotopy theory without $$\mathbb {A}^{1}$$-invariance
Abstract: In this paper, we continue the program initiated by Kahn–Saito–Yamazaki by constructing and studying an unstable motivic homotopy category with modulus $$\overline{\mathbf{M}}\mathcal {H}(k)$$ , 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.