nLab algebraic weak factorization system



Basic idea

Algebraic weak factorization systems (AWFS) are algebraizations of weak factorization systems (WFS). The elements in the left and right classes of morphisms are replaced by coalgebras and algebras, respectively, for a certain comonad and monad on the arrow category. This comonad and monad also determine the functorial factorization and give natural coalgebra and algebra structures to the left and right factors.

Algebraic weak factorization systems were originally called natural weak factorization systems by Grandis and Tholen.


Recall a functorial factorization on a category KK is a functor E:KE : K2KK3 that is a section to the composition functor d 1d_1, induced by the inclusion functor d 1:23d^1 : 2 \rightarrow 3 between the ordinal categories.

Explicitly, EE factors a morphism (u,v):fg(u,v) : f \Rightarrow g in K 2K^{2} as

u f g v E u Lf Lg E(u,v) Rf Rg v \array{ \cdot &\stackrel{u}{\to}& \cdot \\ \downarrow^f && \downarrow^g \\ \cdot &\stackrel{v}{\to}& \cdot } \array{ & & } \stackrel{E}{\mapsto} \array{ &&} \array{ \cdot &\stackrel{u}{\to}& \cdot \\ \downarrow^{Lf} && \downarrow_{Lg} \\ \cdot & \stackrel{E(u,v)}{\to} & \cdot \\ \downarrow^{Rf} && \downarrow_{Rg} \\ \cdot &\stackrel{v}{\to}& \cdot }

There are two other injective functors d 0,d 2:23d^0, d^2 : 2 \rightarrow 3 whose image misses the object that appears as their superscript. When we compose EE with d 2d_2 and d 0d_0, we obtain endofunctors of K 2K^{2}, which we call LL and RR.

There are obvious natural transformations 1R1 \Rightarrow R and L1L \Rightarrow 1 whose components are given by the data of the functorial factorization EE. We say LL and RR are pointed endofunctors, with these natural transformations in mind.


A AWFS on a category KK consists of a pair (L,R)(L,R) where LL is a comonad and RR is a monad, whose underlying pointed endofunctors arise from a functorial factorization EE. Some authors (Garner) also require that the canonical natural transformation LRRLL R \Rightarrow R L, whose domain and codomain components are given by the comultiplication and multiplication maps, is a distributive law of the comonad over the monad. This amounts to the requirement that a pentagon involving the comultiplication and multiplication maps commutes.

We refer to the LL-coalgebras as the left class of the AWFS and the RR-algebras as the right class. When we forget the algebra structures, we obtain classes of maps in KK. The retract closures of these classes form a WFS called the underlying WFS of this AWFS.

Given a lifting problem

u f g v \array{ \cdot &\stackrel{u}{\to}& \cdot \\ \downarrow^f && \downarrow^g \\ \cdot &\stackrel{v}{\to}& \cdot }

where ff is a LL-coalgebra and gg is an RR-algebra, the functorial factorization, coalgebra, and algebra structures can be combined to define a solution, which proves that the left class has the left lifting property with respect to the right class. We leave the details as an exercise.

Interesting features

  • The right class of a AWFS is closed under any limits that exist in K 2K^{2}, because the forgetful functor to the underlying category of arrows creates all limits which exist. Note that it does not follow that the right class of the underlying WFS is closed under limits in the arrow category, because first, it is possible that some elements of the right class will not have an RR-algebra structure, and second, not every map in the arrow category between RR-algebras is necessarily an RR-algebra map.

  • Algebras for the monad of an AWFS can be composed canonically, as can the coalgebras for the comonad. The composition law for the algebras uses the comultiplication natural transformation, and dually for the coalgebras.

  • Each AWFS (L,R)(L, R) on KK induces a right-connected double category RR-𝔸lg\mathbb{A}lg, whose category of objects and horizontal morphisms is KK, whose category of vertical morphisms and cells is RR-AlgAlg.

  • A AWFS (L,R)(L,R) on KK induces a levelwise AWFS on any diagram category K AK^A. Note that its underlying WFS will not be similarly “levelwise”. (Indeed, a WFS does not typically induce a levelwise WFS on a diagram category.)

  • An AWFS can be detected as a functorial factorization that extends to a monad over codcod with a composition law for factorizations.

Small object argument

The algebraic small object argument, an enhancement of the small object argument due to Richard Garner, produces cofibrantly generated AWFS by adapting the construction of a free monad on an endofunctor. Importantly, Garner’s small object argument allows the generators to be a small category over the arrow category K [2]K^{[2]}, rather than simply a set of arrows. As a result, there are WFS which are not cofibrantly generated in the classical sense, but which can be exhibited as the underlying WFS of a cofibrantly generated AWFS.

Every AWFS on a locally presentable category generated by such a small category of maps is in particular an accessible weak factorization system, and every accessible WFS admits an algebraic enhancement generated by the algebraic small object argument. However, not every accessible AWFS is itself generated by the algebraic small object argument, unless we enhance it further to take a double category of generators; see Bourke and Garner.

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.

weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as \toalgebraic small object argument


A new proof of the Strøm model structure using algebraic weak factorization systems:

Introductory texts:

The following dissertation section is entirely written after learning of Riehl (2008), but has complementary examples and may dive deeper into some proofs:

  • Andreas Nuyts, Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems, PhD thesis, KU Leuven, Belgium, 2020

Last revised on May 9, 2023 at 10:44:21. See the history of this page for a list of all contributions to it.