Monads and theories

Given a locally presentable enriched category $\mathcal{E}$ together with a small dense full subcategory $\mathcal{A}$ of arities, we study the relationship between monads on $\mathcal{E}$ and identity-on-objects functors out of $\mathcal{A}$, which we call $\mathcal{A}$-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised as the $\mathcal{A}$-nervous monads—those for which the conclusions of Weber’s nerve theorem hold—and the $\mathcal{A}$-theories, which we introduce here. The resulting equivalence between $\mathcal{A}$-nervous monads and $\mathcal A$-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 $\omega$-groupoids. Besides establishing our general correspondence and illustrating its reach, we study good properties of $\mathcal{A}$-nervous monads and $\mathcal A$-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, Melli`es and Weber.