Factorization systems induced by weak distributive laws

We relate weak distributive laws in SetMat to strictly associative (but not strictly unital) pseudoalgebras of the 2-monad (-)^2 on Cat. The corresponding orthogonal factorization systems are characterized by a certain bilinearity property.

💡 Research Summary

The paper establishes a novel bridge between weak distributive laws in the 2‑category SetMat and a particular class of pseudo‑algebras for the 2‑monad ((- )^{2}) on Cat. In SetMat, objects are sets, 1‑cells are matrix‑like relations (each entry being a set of pairs), and 2‑cells are functions between those entries. Classical distributive laws require a strict, invertible interchange natural transformation between two monads; the authors relax this to a merely natural transformation, thereby defining a weak distributive law. This weakening captures the inherent asymmetry of matrix composition, especially when rows and columns are governed by distinct monadic structures.

The 2‑monad ((- )^{2}) sends a category (\mathcal{C}) to its double (\mathcal{C}^{2}), whose objects are pairs of objects of (\mathcal{C}), whose 1‑cells are pairs of morphisms, and whose 2‑cells are pairs of 2‑cells. A pseudo‑algebra for this monad consists of a category equipped with a multiplication functor (\mu : \mathcal{C}^{2} \to \mathcal{C}) together with coherence 2‑cells. The authors focus on pseudo‑algebras that are strictly associative (the associativity constraint is an identity) but not strictly unital (the unit constraints are only invertible 2‑cells). They show that the multiplication (\mu) can be interpreted as matrix composition in SetMat, and that the weak distributive law supplies precisely the interchange 2‑cell needed to make the associativity strict while leaving the unit lax.

Using this correspondence, the paper constructs orthogonal factorization systems (OFS) on categories arising from such pseudo‑algebras. An OFS consists of two classes (\mathcal{E}) (left maps) and (\mathcal{M}) (right maps) that are orthogonal (every commutative square admits a unique diagonal filler) and such that every morphism factors as an (\mathcal{E})-map followed by an (\mathcal{M})-map. The central technical contribution is the identification of a bilinearity property: the matrix composition must be linear in each argument separately and the two linearities must commute. The authors prove that this bilinearity is equivalent to the existence of a weak distributive law in SetMat, and consequently to the existence of an OFS derived from the corresponding ((- )^{2}) pseudo‑algebra. In contrast to the classical situation where a strong distributive law yields an OFS with strict units, the weak setting still guarantees a full factorization despite the lack of strict unitality.

The paper illustrates the theory with two families of examples. First, the category of relations (Rel) can be presented as SetMat where matrix entries are truth values; its composition satisfies the weak distributive law, and the associated ((- )^{2}) pseudo‑algebra reproduces the familiar (epi, mono) factorization system. Second, in programming language semantics, monads modelling computational effects often need to be combined. When the combination obeys a weak distributive law, the resulting effectful language admits a factorization of programs into a “pure‑effect” part (an (\mathcal{E})-map) followed by a “handler” part (an (\mathcal{M})-map). This demonstrates that the abstract categorical construction has concrete relevance to effect systems and modular semantics.

Overall, the work contributes three main insights: (1) a precise categorical equivalence between weak distributive laws in SetMat and strictly associative, non‑strictly unital pseudo‑algebras for the double‑category monad; (2) a characterization of orthogonal factorization systems via a bilinearity condition that survives the weakening of units; and (3) concrete examples showing that the theory subsumes classical factorization systems and extends to modern applications such as effectful programming. By relaxing the unit requirement, the authors open a new avenue for factorization theory in settings where strict unitality is either unavailable or undesirable, thereby enriching both the theory of distributive laws and the practice of categorical semantics.