Question

A pretty picture for determinant lines comes from algebraic K-theory (

I learned it from Beilinson lecturing on epsilon factorsnow realize this is covered beautifully in the notes by Muro linked to by YBL, but it must be ancient and standard). anyway here's a brief summary: Given a perfect complex of vector spaces (let's work first over a point) we get a homotopy point of the K-theory spectrum, and we can start asking "which point is it" in more and more refined fashion. First we ask for which component it's in (i.e. look at pi_0) - these are labeled by the integers, ie by the Euler characteristic of your complex. Next you can ask to describe it as an object of the fundamental groupoid of K-theory (i.e. give also pi_1 information). This fundamental groupoid is canonically identified with the (Picard) groupoid of graded (super)lines. The grading is given by the Euler characteristic (ie project on pi_0), and the superline is the determinant line of your complex. If you give concrete realizations of the higher fundamental groupoids of the K-theory spectrum you get concrete K-theoretic invariants of your complex of a higher and higher categorical nature (the ultimate one being of course just giving your complex itself as a homotopy point (or contractible subset) of K-theory). You can do the same in families, i.e. over a base, giving a locally constant function on the base from pi_0 (the Euler characteristic of your complex), and a Z-graded super line bundle on the base (determinant line), and so on..

One place this is used beautifully (and where I learned it) is Beilinson's work on epsilon factors.