On Determinant Functors and $K$-Theory

Abstract

We extend Deligne's notion of determinant functor to Waldhausen categories and (strongly) triangulated categories. We construct explicit universal determinant functors in each case, whose target is an algebraic model for the $1$-type of the corresponding $K$-theory spectrum. 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 of (strongly) triangulated categories and obtain generators and (some) relations for various $K_{1}$-groups. This is achieved via a unified theory of determinant functors which can be applied in further contexts, such as derivators.