nLab accessible model category

Definition

An accessible model category is a model structure on a locally presentable category whose two weak factorization systems can be realized by functorial factorizations that are accessible functors, that is that they are accessible weak factorization systems.

This implies that in fact its weak factorization systems can be enhanced to algebraic weak factorization systems that are also accessible (see accessible wfs). However, such algebraic structure is not given as data in the notion of accessible model category: for that, see algebraic model category.

Properties

Left and right lifting

Accessible model structures can be both left- and right-lifted along adjunctions as long as the relevant “acyclicity condition” holds. That is, let $U:A\to B$ be a functor between locally presentable categories and suppose $B$ is an accessible model category. Then:

1. If $U$ is a right adjoint, then there is a model structure on $A$ in which the weak equivalences and fibrations are created by $U$ (i.e. $W_A = U^{-1}(W_B)$ and $F_A = U^{-1}(F_B)$) if and only if every map having the left lifting property with respect to $F_A$ lies in $W_A$.

2. If $U$ is a left adjoint, then there is a model structure on $A$ in which the weak equivalences and cofibrations are created by $U$ (i.e. $W_A = U^{-1}(W_B)$ and $C_A = U^{-1}(C_B)$) if and only if every map having the right lifting property with respect to $C_A$ lies in $W_A$.

Both of the lifted model structures are then again accessible. See transferred model structure. For proofs, see HKRS and its correction in GKR.

Related concepts

• accessible weak factorization system
• algebraic model category

Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.

structure	small-set-generated	small-category-generated	algebraicized
weak factorization system	combinatorial wfs	accessible wfs	algebraic wfs
model category	combinatorial model category	accessible model category	algebraic model category
construction method	small object argument	same as $\to$	algebraic small object argument

References

• Jiri Rosicky, Accessible model categories, Appl Categor Struct (2017) 25: 187. doi, arxiv

• Kathryn Hess, Magdalena Kędziorek, Emily Riehl, Brooke Shipley, A necessary and sufficient condition for induced model structures (arXiv:1509.08154). This paper contains an error, corrected by:

• Richard Garner, Magdalena Kedziorek, Emily Riehl, Lifting accessible model structures, arXiv:1802.09889

• John Bourke, Equipping weak equivalences with algebraic structure, arxiv:1712.02523