2-dimensional bifunctor theorems and distributive laws

TL;DR

This paper unifies bifunctor theorems and distributive laws for pseudofunctors, providing a framework that generalizes and connects these concepts through 2-category theory and lax functors.

Contribution

It introduces a unified framework linking bifunctor theorems and distributive laws via 2-category theory and lax functors, extending the collation process.

Findings

Established a bifunctor theorem for lax functors.

Defined a 2-category of distributive laws equivalent to a lax functor category.

Extended collation to a 2-functor, connecting to uncurrying.

Abstract

In this paper we consider the conditions that need to be satisfied by two families of pseudofunctors with a common codomain for them to be collated into a bifunctor. We observe similarities between these conditions and distributive laws of monads before providing a unified framework from which both of these results may be inferred. We do this by proving a version of the bifunctor theorem for lax functors. We then show that these generalised distributive laws may be arranged into a 2-category Dist(B,C,D), which is equivalent to Lax(B,Lax(C,D)). The collation of a distributive law into its associated bifunctor extends to a 2-functor into Lax($B \times C$, D), which corresponds to uncurrying via the aforementioned equivalence. We also describe subcategories on which collation itself restricts to an equivalence. Finally, we exhibit a number of natural categorical constructions as special cases of our result.