The formal theory of relative monads

TL;DR

This paper develops a comprehensive theory of relative monads and adjunctions within virtual equipments, unifying various concepts and extending classical monad theory to enriched and more general settings.

Contribution

It introduces the theory of relative monads and adjunctions in virtual equipments, providing new universal properties and representation theorems that unify existing concepts.

Findings

Established stronger universal properties for algebra and opalgebra objects.

Proved representation theorems linking various notions of relative monads.

Extended the theory to enriched categories, including the classical case of Set.

Abstract

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the $j$-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment $\mathbb{V}\text{-}\mathbf{\mathbb{C}at}$ of categories enriched in a monoidal category $\mathbb{V}$, though many of our results are new even for $\mathbb{V} = \mathbf{Set}$.