摘要：I will present joint work with Quoc P. Ho on our theory of graded sheaves which provides a uniform construction of mixed versions or graded lifts in the sense of Beilinson--Ginzburg--Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our sheaf theory comes with a six-functor formalism, a perverse t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the six-functor formalism, perverse t-structures, and Frobenius weights on ell-adic sheaves. Historically, constructions of graded lifts were done on a case-by-case basis and were technically subtle, due to Frobenius' non-semisimplicity. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, our categories agree with those previously constructed when they are available. For example, the monoidal DG-category of chain complexes of Soergel bimodules is equivalent to the category of constructible graded sheaves on B\G/B.