First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction.

This was actually forward-referring to Chapter 5, and I am able to read until Chapter 4 inclusively, but Proposition 5.1 knocks me off, despite providing some examples. The short statement of it is

The category L_qcoh (Z/S^1) is equivalent to the category of comodule objects in L_qcoh(Z) for the coalgebra p_∗act^∗ in End(L_qcoh (Z)).

Now since there are some references included, I tried looking there, but I think I just don't "get" something about localization. In fact, I remember some physical examples (Chern-Simons theory, I believe) about localizing the path integral onto the fixed points of some torus -- I'm sure this is relevant here, but I don't know how.

So what I have in mind for this question is that perhaps somebody could provide some simpler examples of localization, or connect it to other places where it naturally arises, e.g. physics.