  In this paper, we show that tropical cycle class maps introduced by Liu for
smooth algebraic varieties over trivially valued fields are isomorphisms. More
precisely, we prove that tropical cohomology is isomorphic to Zariski sheaf
cohomology of the sheaves of tropical analogs of Milnor $K$-groups. Proof is
based on a theorem on "cohomology theories", developed by many mathematicians,
and explicit computation of tropical cohomology of $\mathbb{A}^1$-fibers by
non-archimedean geometry.
