$$\mathbb {A}^1$$-homotopy theory of log schemes

Abstract We construct the $$\mathbb {A}^1$$ A 1 -local stable motivic homotopy categories of fs log schemes. For schemes with the trivial log structure, our construction is equivalent to the original construction of Morel–Voevodsky. We prove that our construction satisfies the localization property. As a consequence, we obtain the Grothendieck six-functor formalism for strict morphisms of fs log schemes. We extend $$\mathbb {A}^1$$ A 1 -invariant cohomology theories of schemes to fs log schemes. In particular, we define motivic cohomology, homotopy K-theory, and algebraic cobordism of fs log schemes. For any fs log scheme log smooth over a scheme, we express cohomology of its boundary in terms of cohomology of schemes.