The theory and practice of Reedy categories

TL;DR

This paper clarifies the role of Reedy categories in homotopy theory, providing simplified proofs of key results and illustrating their applications through examples, making the theory more accessible and easier to apply.

Contribution

It offers a novel proof approach for Reedy category results using weighted colimits and Leibniz constructions, enhancing understanding and accessibility.

Findings

Simplified proofs of Reedy category properties

Validation of homotopy limit and colimit formulas

Examples demonstrating theoretical concepts

Abstract

The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but difficult to find in the literature. While the results presented here are not new, our approach to their proofs is somewhat novel. Specifically, we reduce the much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushout-product) constructions. The general theory is developed in parallel with examples, which we use to prove that familiar formulae for homotopy limits and colimits indeed have the desired properties.