Lifting accessible model structures

TL;DR

This paper establishes conditions under which accessible model structures in locally presentable categories can be lifted along adjoint functors, correcting previous errors and providing a more practical proof for homotopy theory applications.

Contribution

It proves a corrected and more accessible theorem on lifting accessible model structures along adjoints in locally presentable categories.

Findings

Accessible model structures can be lifted along adjoints if an 'acyclicity' condition holds.

The paper corrects a previous incorrect proof and offers a more tractable alternative.

The results apply to a broad class of model structures, including all combinatorial ones.

Abstract

A Quillen model structure is presented by an interacting pair of weak factorization systems. We prove that in the world of locally presentable categories, any weak factorization system with accessible functorial factorizations can be lifted along either a left or a right adjoint. It follows that accessible model structures on locally presentable categories - ones admitting accessible functorial factorizations, a class that includes all combinatorial model structures but others besides - can be lifted along either a left or a right adjoint if and only if an essential "acyclicity" condition holds. A similar result was claimed in a paper of Hess-Kedziorek-Riehl-Shipley, but the proof given there was incorrect. In this note, we explain this error and give a correction, and also provide a new statement and a different proof of the theorem which is more tractable for homotopy-theoretic applications.