nLab composition law for factorizations

Composition laws for factorizations

Composition laws for factorizations


A composition law for a functorial factorization is a functorial way to compose lifting structures for its algebraic right maps. A functorial factorization whose pointed endofunctor extends to a monad over codcod is an algebraic weak factorization system if and only if it has a composition law. Moreover, subject to smallness conditions, any functorial factorization with a composition law freely generates an algebraic weak factorization system.


Suppose 𝒞\mathcal{C} is a category equipped with a functorial factorization, sending every arrow f:ABf:A\to B to a factorization Af LEff RBA \xrightarrow{f_L} E f \xrightarrow{f_R} B. As noted at functorial factorization, a functorial factorization is equivalent to a pointed endofunctor RR on 𝒞 2\mathcal{C}^{\mathbf{2}} over codcod, which maps each morphism ff (regarded as an object of the arrow category 𝒞 2\mathcal{C}^{\mathbf{2}}) to its right factor f Rf_R, the point being given by the left factor f Lf_L and the identity:

f L f f R id .\array{ & \xrightarrow{f_L} & \\ ^f \downarrow & & \downarrow^{f_R} \\ & \xrightarrow{id}&. }

As with any pointed endofunctor, we can consider the category of algebras for RR. Such an RR-algebra is an arrow f:ABf:A\to B equipped with a map s:EfAs : E f \to A such that sf L=id As \circ f_L = id_A and fs=f Rf \circ s = f_R. Equivalently, it is a diagonal lifting in the square

A id A f L f Ef f R B. \array{ A & \xrightarrow{id} & A \\ ^{f_L}\downarrow & & \downarrow^{f} \\ E f & \xrightarrow{f_R} & B.}

In particular, this means that if (L,R)(L,R) is a factorization for a weak factorization system, then the arrows of 𝒞\mathcal{C} that admit some structure of RR-algebra are precisely those in the right class of the weak factorization system.

The morphisms of RR-algebras are commuting squares gh=kfg \circ h = k \circ f that additionally commute with the actions, i.e. hs f=s gE(h,k)h \circ s_f = s_g \circ E(h,k). This defines a category RAlgR Alg with a forgetful functor U:RAlg𝒞 2U : R Alg \to \mathcal{C}^{\mathbf{2}}.

If it should happen that the pointed endofunctor RR is actually a monad over codcod, i.e. it also has a multiplication RRRR R \to R that is also the identity on codomains, then we can also consider the smaller category Alg\mathbb{R} Alg of monad algebras, the RR-algebras as above such that ss also satisfies an associativity condition.


A right weak composition law for a functorial factorization is a functor RAlg× 𝒞RAlgRAlgR Alg \times_{\mathcal{C}} R Alg \to R Alg, where the pullback is over domU:RAlg𝒞dom \circ U : R Alg \to \mathcal{C} and codU:RAlg𝒞cod \circ U : R Alg \to \mathcal{C}, lying over the composition functor 𝒞 2×𝒞 2𝒞\mathcal{C}^{\mathbf{2}} \times \mathcal{C}^{\mathbf{2}}\to \mathcal{C}. If RR is a monad over codcod, then a right strong composition law is defined analogously using Alg\mathbb{R} Alg instead.

More explicitly, this means that

  1. whenever (f,s)(f, s) and (g,t)(g, t) are RR-algebras (resp. \mathbb{R}-algebras) such that cod(f)=dom(g)cod(f) = dom(g), we have a specified RR-algebra structure (resp. \mathbb{R}-algebra structure) tst \bullet s for gfg f, such that
  2. for any morphisms of RR-algebras (resp. \mathbb{R}-algebras) (u,v):(f,s)(f,s)(u, v) : (f, s) \to (f' , s') and (v,w):(g,t)(g,t)(v, w) : (g, t) \to (g' , t') between composable pairs (f,s),(g,t)(f, s), (g, t) and (f,s),(g,t)(f' , s' ),(g' , t' ), the pasted square (u,w):(gf,ts)(gf,ts)(u, w) : (g f, t \bullet s) \to (g ' f' , t' \bullet s ' ) is also a map of RR-algebras (resp. \mathbb{R}-algebras).

In other words, a composition law is an operation with the requisite shape to be the vertical composition in a double category whose vertical arrows are algebras, whose horizontal arrows are arbitrary arrows, and whose 2-cells are commutative squares. Associativity is not assumed, but as noted below it often comes for free.

Relation to awfs


Suppose (L,R)(L,R) is a functorial factorization whose underlying pointed endofunctor RR over codcod has the structure of a monad on 𝒞 2\mathcal{C}^{\mathbf{2}} over codcod. Then (L,R)(L,R) is an algebraic weak factorization system if and only if it admits a right strong composition law.


See Garner 09, Garner 10, Riehl 11, and Barthel-Riehl 13 for proofs in varying degrees of explicitness.


Suppose (L,R)(L,R) is a functorial factorization with a right weak composition law, and that the algebraically free monad on the pointed endofunctor RR exists and is over codcod (for instance if it can be constructed by a transfinite construction of free algebras). Then the latter monad has a right strong composition law, hence underlies an algebraic weak factorization system whose right-monad-algebras coincide with the RR-pointed-endofunctor-algebras, including their natural composition law.


By definition, the algebraically-free monad 𝔽(R)\mathbb{F}(R) satisfies 𝔽(R)Alg=RAlg\mathbb{F}(R) Alg = R Alg. Thus, the weak composition law for RR extends to a strong one for 𝔽(R)\mathbb{F}(R); now we can apply the previous theorem.


  • Richard Garner. Understanding the small object argument. Appl. Categ. Structures. 17(3) (2009) 247–285.

  • Richard Garner. Homomorphisms of higher categories. Adv. Math. 224(6) (2010), 2269–2311.

  • Emily Riehl. Algebraic model structures. New York J. Math. 17 (2011) 173-231.

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

Created on December 30, 2018 at 23:48:46. See the history of this page for a list of all contributions to it.