Constructions of Waldhausen categories via Grothendieck opfibrations

Abstract

Given a Grothendieck opfibration $p:\mathcal{T}\to \mathcal{B}$, we describe a method to construct a Waldhausen category structure on the total category $\mathcal{T}$ via combining Waldhausen category structures on the fibers ${\mathcal{T}}_A$ for $A\in \mathrm{Ob}(\mathcal{B})$ and the basis category $\mathcal{B}$. As an application, we show that if $\mathsf{E}$ is a Waldhausen category with small coproducts such that the class of cofibrations is the left part of a weak factorization system in $\mathsf{E}$, then the representation category $\mathsf{Rep}(Q,\mathsf{coE})$ of a left rooted quiver $Q$ is a Waldhausen category, where $\mathsf{coE}$ is the subcategory of $\mathsf{E}$ whose morphisms are cofibrations.