Constructions of Waldhausen categories via Grothendieck opfibrations

TL;DR

This paper presents a method to construct Waldhausen categories from Grothendieck opfibrations by combining fiberwise and base Waldhausen structures, with applications to representation categories of quivers.

Contribution

It introduces a new construction technique for Waldhausen categories using Grothendieck opfibrations and applies it to quiver representation categories.

Findings

Waldhausen structures can be built on total categories via fibers and bases.

Representation categories of certain quivers are Waldhausen categories.

The method generalizes existing constructions in algebraic K-theory.

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.