The Only Distributive Law Over the Powerset Monad Is the One You Know

TL;DR

The paper investigates when a distributive law of the powerset monad over a set functor exists and is unique by treating extensions to Rel as central objects. It introduces elementwise boundedness to identify a broad class where existence implies Barr extension, and uniqueness follows from preservation of weak pullbacks, with accessible functors obeying these conditions and yielding a canonical extension. A striking contrast is shown: the full powerset functor admits exactly three distinct extensions, illustrating that uniqueness can fail outside the elementwise-bounded/accessible regime. The results clarify when liftings to relations are canonical and connect local monotonicity of extensions to accessibility-like properties. This has implications for relational semantics, coalgebraic logic, and the design of canonical semantics in systems modeling nondeterminism.

Abstract

Distributive laws of set functors over the powerset monad (also known as Kleisli laws for the powerset monad) are well-known to be in one-to-one correspondence with extensions of set functors to functors on the category of sets and relations. We study the question of existence and uniqueness of such distributive laws. Our main result entails that an accessible set functor admits a distributive law over the powerset monad if and only if it preserves weak pullbacks, in which case the so-called power law (which induces the Barr extension) is the unique one. Furthermore, we show that the powerset functor admits exactly three distributive laws over the powerset monad, revealing that uniqueness may fail for non-accessible functors.