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.

Preliminaries

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

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

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

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

There are obvious natural transformations $1 \Rightarrow R$ and $L \Rightarrow 1$ whose components are given by the data of the functorial factorization $E$. We say $L$ and $R$ are pointed endofunctors, with these natural transformations in mind.

Definition

A AWFS on a category $K$ consists of a pair $(L,R)$ where $L$ is a comonad and $R$ is a monad, whose underlying pointed endofunctors arise from a functorial factorization $E$. Some authors (Garner) also require that the canonical natural transformation $L R \Rightarrow R L$, 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 $L$-coalgebras as the left class of the AWFS and the $R$-algebras as the right class. When we forget the algebra structures, we obtain classes of maps in $K$. The retract closures of these classes form a WFS called the underlying WFS of this AWFS.

Given a lifting problem

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

where $f$ is a $L$-coalgebra and $g$ is an $R$-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 $K^{2}$, 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 $R$-algebra structure, and second, not every map in the arrow category between $R$-algebras is necessarily an $R$-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 $(L, R)$ on $K$ induces a right-connected double category $R$-$\mathbb{A}lg$, whose category of objects and horizontal morphisms is $K$, whose category of vertical morphisms and cells is $R$-$Alg$.

• A AWFS $(L,R)$ on $K$ induces a levelwise AWFS on any diagram category $K^A$. 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 $cod$ 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 $K^{[2]}$, 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.

Related concepts

• algebraic model category
• accessible model category
• accessible weak factorization system

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

• Marco Grandis and Walter Tholen, Natural weak factorization systems, Arch. Math. (Brno) 42(4) (2006) 397–408. MR2283020 (2008b:18006), Zbl 1164.18300.

• Richard Garner, Understanding the small object argument, arXiv.

• Emily Riehl, Algebraic model structures, (arXiv:0910.2733).

• Thomas Athorne, The coalgebraic structure of cell complexes, TAC

• Tobias Barthel and Emily Riehl, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. Volume 13, Number 2 (2013), 1089-1124, , arxiv

• John Bourke and Richard Garner, Algebraic weak factorisation systems I: Accessible AWFS, Journal of Pure and Applied Algebra 220, 2016. arXiv:1412.6559, doi:10.1016/j.jpaa.2015.06.002

• John Bourke and Richard Garner, Algebraic weak factorisation systems II: Categories of weak maps, Journal of Pure and Applied Algebra 220, 2016. arXiv:1412.6560, doi:10.1016/j.jpaa.2015.06.003.

• J. Rosicky, Accessible model categories, arxiv

• Emily Riehl, Made-to-Order Weak Factorization Systems, doi, pdf

• Ignacio Lopez Franco, Cofibrantly generated lax orthogonal factorisation systems [arxiv]

• John Bourke, An orthogonal approach to algebraic weak factorisation systems, Journal of Pure and Applied Algebra 227 6 (2023) 107294 [arxiv:2204.09584, doi:10.1016/j.jpaa.2022.107294]

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

• Tobias Barthel, Emily Riehl, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089, euclid:agt/1513715550)

Introductory texts:

• Introduction to Homotopy Theory

• Emily Riehl, Factorization Systems, 2008

• Emily Riehl, §12.4 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

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