Maltsiniotis's first conjecture for K_1

TL;DR

This paper proves that the algebraic K_1 group of an exact category matches that of its associated triangulated derivator, extending to Waldhausen categories with cylinders and saturated weak equivalences.

Contribution

It establishes the equivalence of K_1 groups between certain categories and their derivators, generalizing previous results in algebraic K-theory.

Findings

K_1 of an exact category equals K_1 of its triangulated derivator

K_1 of a Waldhausen category with cylinders matches that of its derivator

extends known K-theory equivalences to broader categorical contexts

Abstract

We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator.