A simple characterization of Quillen adjunctions

TL;DR

This paper identifies a simple criterion under which an enriched right adjoint functor between model categories functions as a right Quillen functor, simplifying the recognition of such adjunctions.

Contribution

It provides a straightforward characterization of Quillen adjunctions based on preservation properties of fibrations and fibrant objects.

Findings

Enriched right adjoint functors preserving acyclic fibrations are often right Quillen functors.

The criterion simplifies the process of identifying Quillen adjunctions in model categories.

The result applies broadly to enriched model categories, streamlining their analysis.

Abstract

We observe that an enriched right adjoint functor between model categories which preserves acyclic fibrations and fibrant objects is quite generically a right Quillen functor.