arXiv2025
We construct a generalization of Morel--Voevodsky's motivic homotopy theory that captures non-$\mathbb{A}^1$-homotopy-invariant phenomena, such as wild ramification and irregular singularity. In the first part, we develop our motivic homotopy theory over quasi-compact and quasi-separated schemes, which satisfies the fundamental properties such as the projective bundle formula, the blow-up sequence, the Gysin sequence, and the Thom isomorphism when the base is normal. Moreover, we compare our theory with existing frameworks. In particular, we recover Morel--Voevodsky's motivic homotopy category and Binda--Park--{\O}stv{\ae}r's logarithmic motivic homotopy category as reflective localizations of our category over normal bases. Furthermore, we construct adjoint functors connecting Annala--Iwasa's category of motivic spectra with ours. In the second part, we equip several non-$\mathbb{A}^1$-homotopy invariant cohomology theories, such as Hodge cohomology, Hodge--Witt cohomology, rank $1$ integrable connections, and unramified cohomology, with canonical filtrations that encode arithmetic and geometric information such as irregular singularities and wild ramification, and prove that these cohomology theories with filtrations are representable in our motivic homotopy category. We also compute some of those filtrations explicitly, and show that they recover known constructions, including a ramification filtration on the Pontryagin dual of the abelian \'etale fundamental group, and an irregularity filtration on the sheaf of rank $1$ connections.