Comparing composites of left and right derived functors

TL;DR

This paper develops a categorical framework using double categories to systematically compare composites of left and right derived functors, enhancing understanding and simplifying proofs in homological algebra.

Contribution

It introduces a novel double categorical approach to compare derived functor composites, generalizing adjunction theory to improve existing proofs.

Findings

Double categories model derived functors with Quillen functors as arrows.

Functorial passage to derived functors is established within this framework.

Applications include improved proofs in homological algebra.

Abstract

We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions and mates in 2-categories, then gives us canonical ways to compare composites of left and right derived functors. We give a number of sample applications, most of which are improvements of existing proofs in the literature.