I now realize this is covered beautifully in the notes by Muro linked to by YBL, but 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.

I now realize this is covered beautifully in the notes by Muro linked to by YBL, but 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.

A pretty picture for determinant lines comes from algebraic K-theory (I learned it from Beilinson lecturing on epsilon factors but it must be ancient and standard). 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..

I now realize this is covered beautifully in the notes by Muro linked to by YBL, but 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.