nLab accessible weak factorization system

Definition

A weak factorization system on a locally presentable category is accessible if it admits a functorial factorization that is an accessible functor.

Equivalent forms

Theorem

For a weak factorization system $(L,R)$ on a locally presentable category $M$, the following are equivalent.

1. $(L,R)$ is accessible.
2. There is a small category $J \to M^\to$ over the arrow category such that $R$ consists of the morphisms with coherent right lifting functions relative to $J$.
3. $(L,R)$ can be generated by the algebraic small object argument.
4. $(L,R)$ can be equipped with the structure of an accessible algebraic weak factorization system.

Proof

Properties of the algebraic small object argument yield (2)$\Rightarrow$(3)$\Rightarrow$(4), and (4)$\Rightarrow$(1) since an algebraic wfs is in particular a functorial factorization. The remaining implication is the following lemma, which is Remark 3.1.8 of HKRS15, which relies on Theorem 4.3 of Rosicky17.

Lemma

Suppose $E$ is an accessible functorial factorization on a locally presentable category $M$, realizing a weak factorization system. Then there is an accessible algebraic weak factorization system realizing the same weak factorization system.

Proof

Let $Coalg(L)$ be the category of coalgebras for the copointed endofunctor of the arrow category $M^\to$ induced by $E$, i.e. morphisms equipped with a section of their $E$-factorization exhibiting them as a retract of the first factor. Then $Coalg(L)$ is locally presentable (being complete and a PIE limit construction from $M$). Thus, it has a small dense subcategory $X$. We can then apply Garner's small object argument to generate an algebraically-free algebraic weak factorization system from $X$. The algebraic right-maps in this awfs are the morphisms with coherent lifting functions against $X$, which by density is the same as having a coherent lifting function against all of $Coalg(L)$, which is the same as being an algebra for the pointed endofunctor of $M^\to$ corresponding to $E$. Thus this is an awfs with the same right-maps, hence the same underlying weak factorization system.

Regarding point (4), note that being accessible is a property of a wfs, while being algebraic is a structure on it. A given accessible wfs can admit many different algebraic realizations, and not all of them (even the accessible ones) may be produced by the ordinary algebraic small object argument (although they can all be produced by the fancier version involving a double category of maps as input).

Properties

Left and right lifting

Any accessible weak factorization system can be right-lifted along a right adjoint, or also left-lifted along a left adjoint, between locally presentable categories. That is, if $U:A\to B$ is a functor between locally presentable categories and $(L,R)$ is an accessible weak factorization system of $B$, then:

1. If $U$ is a right adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $R' = U^{-1}(R)$.

2. If $U$ is a left adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $L' = U^{-1}(L)$.

See HKRS and its correction in GKR for details. In particular, this is useful for the construction of transferred model structures.

Related concepts

• algebraic weak factorization system
• accessible model structure

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