On modified Reedy and modified projective model structures

TL;DR

This paper introduces variations of Reedy and projective model structures that enable selective entry consideration and different model categories at diagram entries, leading to a bisimplicial model category for algebraic K-theory recovery.

Contribution

It proposes new modified model structures on diagram categories, facilitating flexible weak equivalence definitions and diverse model categories at diagram entries.

Findings

Developed modified Reedy and projective model structures.

Constructed a bisimplicial model category for algebraic K-theory.

Enabled recovery of algebraic K-theory for Waldhausen subcategories.

Abstract

Variations on the notions of Reedy model structures and projective model structures on categories of diagrams in a model category are introduced. These allow one to choose only a subset of the entries when defining weak equivalences, or to use different model categories at different entries of the diagrams. As a result, a bisimplicial model category that can be used to recover the algebraic K-theory for any Waldhausen subcategory of a model category is produced.