Motivic homotopy theory for perfect schemes

We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by $p$ is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an $\mathbb{F}_p$-scheme $S$ to the category $\mathrm{S}\mathcal{H}(S)[1/p]$ is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of $\mathrm{S}\mathcal{H}$ at the universal homeomorphisms, which we conclude is the same as $\mathrm{S}\mathcal{H}[1/p]$.