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

Abstract: In this short note, for a morphism of Waldhausen categories $f\colon \mathbb{A} = (\mathcal{A} ,w_{\mathbb{A}}) \to \mathbb{B} = (\mathcal{B},w_{\mathbb{B}})$, we will define $\operatorname{Cone} f$ to be a Waldhausen category. There exists the canonical morphism of Waldhausen categories $\kappa_f\colon \mathbb{B}\to \operatorname{Cone} f$. We will show that the sequence $\mathbb{A}\overset{f}{\to}\mathbb{B}\overset{\kappa_f}{\to}\operatorname{Cone}f$ induces fibration sequence of spaces $K(\mathbb{A})\overset{K(f)}{\to}K(\mathbb{B})\overset{K(\kappa_f)}{\to} K(\operatorname{Cone} f)$ 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.