Abstract: I will exhibit connections between:

1) The Knudsen-Mumford theorem on the existence of a unique determinant functor from bounded complexes to invertible sheaves.

2) Deligne's Picard category of virtual objects of an exact category, which models the 1-truncation of its K-theory spectrum.

3) The Gillet-Waldhausen isomorphism between the K-theories of an exact category and its category of bounded complexes.

If time allows I will also show connections with:

4) Determinant functors on triangulated categories with a "triangulated category of true triangles", as suggested by Grothendieck.

5) Maltsiniotis's conjecture on the agreement between the K-theories of an exact category and its triangulated derivator.