Stone Duality for Monads

Abstract: In computer science, monads are often interpreted as representing computation which interact with some resource or environment. To what extent can we interpret arbitrary monads in this way?

To answer this question, I will associate to each monad its topological behaviour category, whose objects are understood as states of the environment, and whose morphisms are transitions along states. I will also explain how to recover a monad from a topological category, giving rise to an adjunction between the category of monads on set and a category of topological categories.

We interpret this as a Stone-type adjunction because any Boolean algebra induces a monad of "if-then-else" programs, whose behaviour category has space of objects the Stone spectrum, and the only morphisms are identity. If time permits, I will also explain the relationship between our behaviour category and the Zariski spectrum of commutative rings, using an abstract framework of spectra invented by Diers and Cole.

This is joint work with Richard Garner.

computer science theoryMathematics

Audience: researchers in the topic

University of Birmingham theoretical computer science seminar

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