On determinant functors and $K$-theory

TL;DR

This paper develops a new approach to determinant functors that extends their applicability to various categories and connects them with low-dimensional K-theory, answering open questions and establishing new additivity results.

Contribution

It introduces a universal construction of determinant functors for broader categories and links them to K-theory in low dimensions, solving open problems.

Findings

Universal determinant functors constructed for Waldhausen, triangulated categories, and derivators.

Established that these functors compute K-theory in dimensions 0 and 1.

Proved additivity theorems and provided generators and relations for K_1-groups.

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.