Lifting accessible model structures

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–Kędziorek–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.

Injective and projective model structures on enriched diagram categories

“…By results in [GKR18], a weak factorization system can be lifted along the left or right adjoint of an adjunction when it is accessible.…”

“…locally presentable model categories with accessible functorial factorizations. In this paper, we provide an enriched version of this result using the methods of lifting model structures of [HKRS17] and [GKR18]. We prove the following main theorem.…”

“…In the non-enriched case, the existence of such model structures on categories of diagrams in a combinatorial model category is folklore. However, the breakthrough by Garner, Hess, Kędziorek, Riehl, and Shipley in [HKRS17], and then in [GKR18], provides new tools which unify the proof of existence of both injective and projective model structures under weaker assumptions, namely for accessible model categories, i.e. locally presentable model categories with accessible functorial factorizations.…”

“…The non-enriched case. Before presenting an outline of our arguments in the enriched case, let us explain how the proof of the existence of the injective and projective model structures in the non-enriched case works, using the methods of lifting model structures of [HKRS17] and [GKR18]. This will help us to highlight the similarities, but also the key differences between the two situations.…”

“…In Section 2, following [BQR98] and [Kel82b], we introduce the notions of a locally presentable base V, and of enriched local presentability for a V-category A, which implies the local presentability (in the non-enriched sense) of the category of enriched diagrams D → A, where D is a small V-category. In Section 3, we recall the notion of an accessible model category, and state the Acyclicity Theorem from [GKR18]. In Section 4, we construct an enriched category of objects for a small V-category D, which plays a similar role to the discrete category of objects above, but in the enriched setting.…”