Delooping of the $K$-theory of strictly derivable Waldhausen categories

Abstract

In this short note, for a morphism of Waldhausen categories $f:\mathbb{A}=(\mathcal{A},w_{\mathbb{A}})\to \mathbb{B}=(\mathcal{B},w_{\mathbb{B}})$, we will define $\mathrm{Cone}f$ to be a Waldhausen category. There exists the canonical morphism of Waldhausen categories ${\kappa }_f:\mathbb{B}\to \mathrm{Cone}f$. We will show that the sequence $\mathbb{A}\stackrel{f}{\to }\mathbb{B}\stackrel{{\kappa }_f}{\to }\mathrm{Cone}f$ induces fibration sequence of spaces $K(\mathbb{A})\stackrel{K(f)}{\to }K(\mathbb{B})\stackrel{K({\kappa }_f)}{\to }K(\mathrm{Cone}f)$ on connective $K$-theory. Moreover we will define a notion of strictly derivable Waldhausen categories and define non-connective $K$-theory for strictly derivable Waldhausen categories.