nLab composition law for factorizations

Idea

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.

Definition

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:

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

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

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

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.

Definition

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

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

Theorem

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.

Proof

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

Theorem

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.

Proof

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.

References

• 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