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

2026-02-13

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Blockchains and distributed ledgers
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• distributive law
• powerset monad
• set functor
• weak pullback preservation
• uniqueness

Innovation

• distributive law
• powerset monad
• accessible functor
• weak pullback preservation
• Barr extension