# Techniques: Localisable Monads

## What the paper contributes (one paragraph)
This is a category-theory paper about the semantics of computational side-effects. Monads model effects in functional programming, and the usual way to combine several effects is "bottom-up" via distributive laws, indexed monads, or graded monads. The authors instead take a "top-down" view using tensor topology: any symmetric monoidal category decomposes as a sheaf of local monoidal categories over a base space of central idempotents (its "open sets"). They define a localisable monad as one that respects this decomposition, so that it corresponds to a sheaf of monads on the local categories, and prove a bijective correspondence between localisable monads on a stiff category and formal monads in a presheaf 2-category, then characterise the resulting Eilenberg-Moore and Kleisli algebras. Three illustrative examples interpret the base space as memory locations (a state monad, replicated in Hilbert spaces as a "quantum buffer" of unentangled qubits), as agents in a concurrent network, and as time in stochastic processes.

## Techniques offered
- **Sheaf-theoretic decomposition of monads (localisable monads via tensor topology)** - a categorical framework that decomposes a monad on a monoidal category into a sheaf of local monads over its space of central idempotents, equivalently a formal monad in a presheaf 2-category.
  - guarantee: a representation/correspondence theorem (bijective correspondence between localisable monads on a stiff category and formal monads on the associated presheaf), not a quantitative or operational assurance about any computation
  - quote: "We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads."

## Where it could apply
- Target primitive(s): none
- Target application group(s): none (categorical semantics / programming-language theory; the quantum buffer is an illustrative model, not platforms_tooling)

## Caveats
This is a pure category-theory and programming-semantics paper with nothing suppliable to an external quantum vendor or application result. The single "quantum" content is an illustrative example showing the state monad can be replicated in the category of Hilbert spaces ("this monad models a quantum buffer of $n$ qubits, because there is no entanglement between the different qubits"); it is a categorical model of unentangled qubit storage, not error correction, decoding, compilation, mitigation, verification, characterisation, simulation, or any algorithm that could be applied to a quantum computation. The one bullet above records the genuine contribution for completeness, but its supply to any external result is effectively nil.