[Paper Review] Stone Duality for Monads

Richard Garner, Alyssa Renata+1 more|arXiv (Cornell University)|Mar 26, 2026

Homotopy and Cohomology in Algebraic Topology0 citations

TL;DR

This paper establishes a contravariant idempotent adjunction between finitary/infinitary monads on Set and internal/localic categories with retrofunctors, and identifies fixed points as hyperaffine-unary monads and ample localic categories, yielding a Stone-type duality for monads.

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.

Motivation & Objective

• Motivate a non-shadowed description of computation beyond monad equations by using comodels and behavior categories.
• Introduce a topological/localic framework to capture the universal transition system of a monad.
• Extend from finitary to infinitary monads via localic behaviour categories and ample localic categories.
• Characterize fixed points of the adjunction as hyperaffine-unary monads and ample localic categories.
• Relate the duality to Stone duality by restricting to Boolean algebras and their associated monads.

Proposed method

• Construct the topological behaviour category BT for finitary monads and its functorial extension B: Mnd_ω(Set) → TopRetro^op.
• Use Gamma_ω as the right adjoint to B to obtain the first adjunction between monads and topological retrofunctors.
• Generalize to infinitary monads by replacing TopRetro with LocRetro and defining the localic behaviour category LB T.
• Define LB_0 T as the terminal localic comodel and LB_1 T via a sheaf over LB_0 T to form the localic behaviour category LB T.
• Show LB T and the functor LB give an adjunction with Gamma that preserves sections and yields a Stone duality when restricted.
• Characterize fixed points of the adjunction as cartesian closed (hyperaffine-unary) monads and ample localic categories (localic categories with local homeomorphism source maps and strongly zero-dimensional object locales).

Experimental results

Research questions

• RQ1 How can monads be characterized by an adjunction with internal/localic categories capturing computation as interaction?
• RQ2 What are the fixed points of the adjunction between monads and internal/localic behaviour categories, and how do they relate to known notions like hyperaffine-unary monads and ample localic categories?
• RQ3 Can Stone duality be extended from Boolean algebras to a broad duality between monads and localic categories?
• RQ4 How do finitary versus infinitary monads behave under the LB/Γ adjunction and its finite-information topology?
• RQ5 In what sense does the localic approach recover or approximate the original monad for various theories of computation?

Key findings

• There exists a contravariant idempotent adjunction between ranked monads on Set and internal categories with internal retrofunctors in Loc, realized via LB T and Γ LB T.
• The left adjoint maps a monad T to its localic behaviour category LB T, a universal transition system for interactions with T.
• The right adjoint maps a localic category LC to the monad Γ LC, defined by A-indexed families of local sections to the source map that partition the object locale.
• Fixed points of the adjunction consist of hyperaffine-unary monads and ample localic categories, yielding Stone duality for monads.
• Hyperaffine-unary monads correspond to cartesian closed Eilenberg-Moore categories, and ample localic categories have local homeomorphism sources with strongly zero-dimensional object locales.
• Restriction to Stone spaces and Boolean algebras recovers the classical Stone duality as a special case.