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Stream: deprecated: our papers

Topic: The formal theory of relative monads

Nathanael Arkor (Feb 28 2023 at 14:21):

@Dylan McDermott and I have a new preprint on arXiv: The formal theory of relative monads.

It is the first in a series of papers on the theory of [[relative monads]]. Here's the 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 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 $\mathbb V = \mathbf{Set}$.

We hope the paper is of interest to a range of readers.

• For readers interested purely in relative monads (e.g. ordinary or enriched), we have a number of new results. For instance, we show that every relative monad may be viewed as a monoid in a suitable skew-multicategory (thereby dropping the requirement of Altenkirch–Chapman–Uustalu that enough pointwise left extensions exist). The introduction lists some of the new results about relative monads, allowing you to skip over the formal category theory if you're not interested in that aspect. We intended that at least the theorem statements, and most of the discussion, are readable to those who are only familiar with ordinary or enriched category theory.
• For readers interested in the formal theory of monads in a 2-category, there are several aspects of the formal theory of relative monads that behave differently from the non-relative setting. For one thing, it is necessary to work in richer setting than a 2-category (we work in a [[virtual equipment]]). Another is that stronger universal properties are required of the algebra- and opalgebra-objects to recover expected theorems (namely that (op)algebra-objects induce (op)monadic relative adjunctions).
• For readers interested in formal category theory more generally, we develop some basic formal category theory in a virtual equipment, which should be useful more generally. Most of the results are well-known in other contexts (e.g. in [[proarrow equipments]]), but had not yet been generalised to this context.

We're very happy to discuss the paper, and answer any questions anyone has about it :)