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 qua trivial relative monad are stronger than the classical notions of algebra object and opalgebra object for a monad qua monad. 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 {Cat} of categories enriched in a monoidal category \mathbb {V}, though many of our results are new even for \mathbb {V}=\mathbf {Set}.
@misc{arkor-mcdermott-2023-formal,
title={The formal theory of relative monads},
author={Nathanael Arkor and Dylan McDermott},
year={2023},
eprint={2302.14014},
archivePrefix={arXiv},
primaryClass={math.CT}
}