Delooping of the $K$-theory of Waldhausen categories with factorizations

In this article, we will provide new delooping methods of the $K$-theory of certain Waldhausen categories and abelian categories. A specific feature of our delooping is that a suspension of an abelian category is again an abelian category. By utilizing this delooping method, we will show that negative K-groups of an abelian category is trivial which is conjectured by Marco Schlichting and as its consequence, we will obtain a generalization of a theorem of Auslander and Sherman which says that negative direct sum $K$-groups and negative $K$-groups are isomorphisms for any small exact categories.