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

TL;DR

This paper investigates the conditions under which set functors admit distributive laws over the powerset monad, establishing that preservation of weak pullbacks ensures a unique such law, with implications for the structure of these laws.

Contribution

It proves that accessible set functors preserve weak pullbacks and have a unique distributive law over the powerset monad, identifying exactly three such laws for the powerset functor.

Findings

Accessible set functors admit a unique distributive law if they preserve weak pullbacks.

The power law is the unique distributive law for accessible functors.

Non-accessible functors may have multiple distributive laws, showing non-uniqueness.

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.