Grothendieck topologies with logarithmic modifications

Many concepts in logarithmic geometry are invariant under log blowups. To formalize this invariance, we introduce the m-open, m-\'etale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes. These refine the standard topologies from scheme theory by treating log modifications as covers. In constructing them, we identify and correct errors in the definitions of log modifications and the full log \'etale topology. Our m-topologies are variants of those introduced by Niziol and Park; specifically, the m-\'etale topology is a subtopology of Kato's full log \'etale topology, characterized by a stronger lifting property than for log \'etale maps. This strengthening ensures the functoriality of the corresponding small site. We also characterize the sheaves for all these sites and connect the m-open site to Kato's valuative space.