Monads and theories

Abstract

Given a locally presentable enriched category E together with a small dense full subcategory A of arities, we study the relationship between monads on E and identity-on-objects functors out of A, which we call A-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised on the one side as the A-nervous monads—those for which the conclusions of Weber's nerve theorem hold—and on the other, as the A-theories which we introduce here.The resulting equivalence between A-nervous monads and A-theories is best possible in a precise sense, and extends almost all previously known monad–theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak ω-groupoids.Besides establishing our general correspondence and illustrating its reach, we study good properties of A-nervous monads and A-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Melliès and Weber.