Localisable Monads
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Monads govern computational side-effects in programming semantics. They can be combined in a ”bottom-up” way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we equip monads with fine-grained structure in a ”top-down” way, using techniques from tensor topology. This provides an intrinsic theory of local computational effects without needing to know how constituent effects interact beforehand. Specifically, any monoidal category decomposes as a sheaf of local categories over a base space. We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads. Equivalently, localisable monads are formal monads in an appropriate presheaf 2-category, whose algebras we characterise. Three extended examples demonstrate how localisable monads can interpret the base space as locations in a computer memory, as sites in a network of interacting agents acting concurrently, and as time in stochastic processes.
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