Triangulated Categories of Logarithmic Motives Over a Field

fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and
the basic idea of parameterizing homotopies by □, i.e., the projective line with respect
to its compactifying logarithmic structure at infinity. We show that Hodge cohomology
of log schemes is a □-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voevodsky's category of motives
and A1-invariant theories: assuming resolution of singularities, we identify the latter
with the full subcategory comprised of A1-local objects in the category of logarithmic
motives. Fundamental properties such as □-homotopy invariance, Mayer-Vietoris for
coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well
as a blow-up formula and a projective bundle formula witness the robustness of the
setup

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