It is useful to recap what I did today, or these days.

I have been thinking about Koszul duality a lot these days.

• where does the mysterious grading come from. What is mixed category? The terminology is weird that, mixed constructible sheaf does not form a mixed category. you need to take a (not full) subcategory.
• why the two spaces, Coulomb branch and Higgs branch, are related? I don't see the relation at all. You can say, categorically, one is about $Map(S^1, V/G)_{dR}$ the other is $Map(S^1_{dR}, V/G)$. But concretely, on a low brow way, what are we talking about?

You want to say, for the same theory, the Coulomb branch and Higgs branch, are just two shadows of the same object, the theory. Therefore, the category of line operators in the two branches are the same.