# 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.

## Key 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 $\mathbb{V} = \mathbf{Set}$.

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Source: https://tomesphere.com/paper/2302.14014