arXiv:2603.25710 · v4

Stone Duality for Monads

首次提交 2026-03-27 01:53

版本更新 2026-11-02 09:47

雷达观测 2026-11-04 23:42

摘要

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.

作者备注：19 pages without appendix, 35 pages total, to appear in final proceedings of MFPS2026