Filtered modules over filtered sheaves of rings appear naturally in mathematics, such as for example when studying \(\mathscr{D}_X\)-modules on a complex manifold \(X\), \(\mathscr{D}_X\)-denoting the filtered ring of differential operators (See [M. Kashiwara, \(D\)-modules and microlocal calculus. Translated from the Japanese by Mutsumi Saito. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.32012)]). In this paper, the authors prove that for an abelian category \(\mathscr{C}\) and a filtered preordered set \(\Lambda\), the derived category of the quasi-abelian category of filtered objects in \(\mathscr{C}\) indexed by \(\Lambda\) is equivalent to the derived category of the abelian category of functors from \(\Lambda\) to \(\mathscr{C}\). Also, they apply this result to the study of the category of filtered modules over a filtered ring in a tensor category.
For the entire collection see [Zbl 1353.32001].