K-flat complexes of abelian sheaves in the bounded-above setting

Let $X$ be a topological space. A bounded-above cochain complex ${\mathcal{K}}^{\bullet }$ of abelian sheaves on $X$ (Cochain complex in an abelian category, Bounded, bounded below, and bounded above complexes) is K-flat in the bounded-above sense used on this page when for every acyclic bounded-above complex ${\mathcal{F}}^{\bullet }$ of abelian sheaves, acyclicity being exactness in every degree (Exactness of a complex at a degree and acyclic complexes, Cohomology object of a cochain complex), the tensor-product total complex $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ of Tensor product of abelian sheaves and its total complex is acyclic.

This is the bounded-above variant of the K-flat complexes of the Stacks Project (Definition 26.2), restricted to the complexes between which the tensor-product total complex of Tensor product of abelian sheaves and its total complex is defined; no unbounded total complex is used on this page. The two facts that make the notion useful are proved on this page: a bounded-above complex of flat sheaves (Flat abelian sheaves) is K-flat (Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes), and tensoring with a K-flat complex preserves quasi-isomorphisms (Quasi-isomorphism, K-flat sheaf complexes preserve quasi-isomorphisms).