Stone Duality for Monads

2026-03-26

AI Summary

This work establishes a Stone-type duality between ranked monads on sets and localic categories viewed as behavioral systems. By constructing a contravariant idempotent adjunction, it relates ranked monads to localic categories equipped with an internal inverse functor, characterizing their universal transition systems via behavioral categories. The central contribution is the first explicit duality between hyperaffine–unary monads and strongly zero-dimensional localic categories whose source maps are local homeomorphisms, with classical Stone duality emerging as a special case. This result extends the dual-theoretic foundations of computational semantics, employing tools from category theory, locale theory, internal categories, and Eilenberg–Moore algebras.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Behavioral Game Theory

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce a contravariant idempotent adjunction between (i) the category of ranked monads on $\mathsf{Set}$; and (ii) the category of internal categories and internal retrofunctors in the category of locales. The left adjoint takes a monad $T$-viewed as a notion of computation, following Moggi-to its localic behaviour category $\mathsf{LB}T$. This behaviour category is understood as "the universal transition system" for interacting with $T$: its "objects" are states and the "morphisms" are transitions. On the other hand, the right adjoint takes a localic category $\mathsf{LC}$-similarly understood as a transition system-to the monad $Γ\mathsf{LC}$ where $(Γ\mathsf{LC})A$ is the set of $A$-indexed families of local sections to the source map which jointly partition the locale of objects. The fixed points of this adjunction consist of (i) hyperaffine-unary monads, i.e., those monads where term $t$ admits a read-only operation $\bar{t}$ predicting the output of $t$; and (ii) ample localic categories, i.e., whose source maps are local homeomorphisms and whose locale of objects are strongly zero-dimensional. The hyperaffine-unary monads arise in earlier works by Johnstone and Garner as a syntactic characterization of those monads with Cartesian closed Eilenberg-Moore categories. This equivalence is the Stone duality for monads; so-called because it further restricts to the classical Stone duality by viewing a Boolean algebra $B$ as a monad of $B$-partitions and the corresponding Stone space as a localic category with only identity morphisms.

Problem

• Stone duality
• monads
• localic categories
• behaviour category
• ranked monads

Innovation

• Stone duality
• monads
• localic categories
• hyperaffine-unary monads
• idempotent adjunction