Enhancing the filtered derived category

The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical algebra, in the formalisms of stable ∞-categories, stable model categories, and pretriangulated, idempotent-complete dg categories. We characterize the filtered stable ∞-category Fil(C) of a stable ∞-category C as the left exact localization of sequences in C along the ∞-categorical version of completion (and prove analogous model and dg category statements). We also spell out how these constructions interact with spectral sequences and monoidal structures. As examples of this machinery, we construct a stable model category of filtered D-modules and develop the rudiments of a theory of filtered operads and filtered algebras over operads. This paper is also available at arXiv:1602.01515v3.