On the 3-arrow calculus for homotopy categories

TL;DR

This paper introduces a 3-arrow calculus for localising categories, simplifying the process of representing morphisms in homotopy categories without requiring functorial factorizations.

Contribution

It develops a new 3-arrow calculus framework for localisation, applicable to model categories and derived categories, avoiding zigzags of arbitrary length and functorial factorizations.

Findings

Provides a 3-arrow calculus for homotopy categories

Applicable to localisations of model categories and derived categories

Simplifies localisation without functorial factorizations

Abstract

We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. The methods to construct this localisation are similar to the Ore localisation for a 2-arrow calculus; in particular, we do not have to use zigzags of arbitrary length. Applications include the localisation of an arbitrary model category with respect to its weak equivalences as well as the localisation of its full subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy category in all four cases. In contrast to the approach of Dwyer, Hirschhorn, Kan and Smith, the model category under consideration does not need to admit functorial factorisations. Moreover, our method shows that the derived category of any abelian (or idempotent splitting exact) category admits a 3-arrow calculus if we localise the category of complexes instead of its homotopy category.