Grothendieck topologies with logarithmic modifications

Many concepts in log geometry are invariant under log blow-ups.

To formalize this invariance, we introduce the m-open, m-étale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes.

These refine the standard topologies from scheme theory by treating abstract log modifications as covers.

For example, the m-étale topology is a subtopology of full log étale topology, characterized by a stronger lifting property than for log étale maps.

Along the way, we identify and correct an error in the definition of the full log étale topology.

We also prove a global integralization theorem by logarithmic blow-ups and use it to describe the m-open topos as a limit over log blow-ups.

Finally, we characterize the sheaves on the m-type sites and connect the m-open site to Kato's valuative space.