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

We construct the $\mathbb{A}^1$-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 the localization property.

As a consequence, we obtain the Grothendieck six functors formalism for strict morphisms of fs log schemes.

We extend $\mathbb{A}^1$-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.