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Nathanael Arkor (Aug 26 2025 at 09:11):@Andrew Slattery and I have a new preprint on arXiv: Idempotence for relative monads. Here's the abstract:
We study the concept of idempotence for relative monads, which exhibits several subtleties not present for non-relative monads. In particular, there is a bifurcation of notions of idempotence in the relative setting, which are indistinguishable for idempotent monads. As a special case, we obtain several characterisations of idempotence for monads in extension form.
Idempotent monads are well studied in category theory. They have many equivalent characterisations and lots of useful properties. This paper is a short study on the extent to which these characterisations and properties extend to [[relative monads]]. It turns out that there are two natural notions of idempotent relative monad (both of which specialise to idempotent monads), and we study the relationship between them. This work is an outgrowth of our earlier work with @Philip Saville on pseudoalgebras for (lax-idempotent) relative pseudomonads, which is where we first observed some of these phenomena.