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 Fₚ-scheme S to the category SH(S)[1/p] is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of SH at the universal homeomorphisms, which we conclude is the same as SH[1/p].