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 $cod$ 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 $\mathcal{C}$ is a category equipped with a functorial factorization, sending every arrow $f:A\to B$ to a factorization $A \xrightarrow{f_L} E f \xrightarrow{f_R} B$. As noted at functorial factorization, a functorial factorization is equivalent to a pointed endofunctor $R$ on $\mathcal{C}^{\mathbf{2}}$ over $cod$, which maps each morphism $f$ (regarded as an object of the arrow category $\mathcal{C}^{\mathbf{2}}$) to its right factor $f_R$, the point being given by the left factor $f_L$ and the identity:
As with any pointed endofunctor, we can consider the category of algebras for $R$. Such an $R$-algebra is an arrow $f:A\to B$ equipped with a map $s : E f \to A$ such that $s \circ f_L = id_A$ and $f \circ s = f_R$. Equivalently, it is a diagonal lifting in the square
In particular, this means that if $(L,R)$ is a factorization for a weak factorization system, then the arrows of $\mathcal{C}$ that admit some structure of $R$-algebra are precisely those in the right class of the weak factorization system.
The morphisms of $R$-algebras are commuting squares $g \circ h = k \circ f$ that additionally commute with the actions, i.e. $h \circ s_f = s_g \circ E(h,k)$. This defines a category $R Alg$ with a forgetful functor $U : R Alg \to \mathcal{C}^{\mathbf{2}}$.
If it should happen that the pointed endofunctor $R$ is actually a monad over $cod$, i.e. it also has a multiplication $R R \to R$ that is also the identity on codomains, then we can also consider the smaller category $\mathbb{R} Alg$ of monad algebras, the $R$-algebras as above such that $s$ also satisfies an associativity condition.
A right weak composition law for a functorial factorization is a functor $R Alg \times_{\mathcal{C}} R Alg \to R Alg$, where the pullback is over $dom \circ U : R Alg \to \mathcal{C}$ and $cod \circ U : R Alg \to \mathcal{C}$, lying over the composition functor $\mathcal{C}^{\mathbf{2}} \times \mathcal{C}^{\mathbf{2}}\to \mathcal{C}$. If $R$ is a monad over $cod$, then a right strong composition law is defined analogously using $\mathbb{R} Alg$ instead.
More explicitly, this means that
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.
Suppose $(L,R)$ is a functorial factorization whose underlying pointed endofunctor $R$ over $cod$ has the structure of a monad on $\mathcal{C}^{\mathbf{2}}$ over $cod$. Then $(L,R)$ 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 $(L,R)$ is a functorial factorization with a right weak composition law, and that the algebraically free monad on the pointed endofunctor $R$ exists and is over $cod$ (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 $R$-pointed-endofunctor-algebras, including their natural composition law.
By definition, the algebraically-free monad $\mathbb{F}(R)$ satisfies $\mathbb{F}(R) Alg = R Alg$. Thus, the weak composition law for $R$ extends to a strong one for $\mathbb{F}(R)$; 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 18:48:46. See the history of this page for a list of all contributions to it.