Higher Theories and Monads

TL;DR

This paper extends classical monad-theory correspondences to the realm of $at$-categories, proving new results about the structure and colimits of monads and their algebras in this higher categorical setting.

Contribution

It generalizes the adjunction between monads and pretheories to $at$-categories, providing new tools for constructing and understanding monads in higher categories.

Findings

Category of algebras for accessible monads is locally presentable

Diagrams of accessible monads admit colimits

Provides a simpler construction of monads via theories

Abstract

We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.