On determinant functors and $K$-theory

Abstract

In this paper we introduce a new approach to determinant functors which allows us to extend Deligne's determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universal determinant functor computes the corresponding $K$-theory in dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional $K$-theory and obtain generators and (some) relations for various $K_{1}$-groups.Comment: 73 pages. We have deeply revised the paper to make it more accessible, it contains now explicit examples and computations. We have removed the part on localization, it was correct but we didn't want to make the paper longer and we thought this part was the less interesting one. Nevertheless it will remain here in the arXiv, in version 1. If you need it in your research, please let us kno