On the Centre of Strong Graded Monads
Flavien Breuvart, Quan Long, Vladimir Zamdzhiev
TL;DR
The paper extends the concept of the centre from ungraded strong monads to pomonoid-graded strong monads, showing that a existing centre induces a pomonoid-graded commutative submonad ${\mathcal{Z}}$ of the original monad ${\mathcal{T}}$ when centralisable. It develops graded notions of strength, costrength, and commutativity, and defines morphisms between strong graded monads via monoid homomorphisms between gradings. The centre is lifted to the graded setting as a terminal graded central cone, yielding a graded commutative submonad with monomorphisms into the original graded monad; this is demonstrated via a graded analogue of central cones and an extended proof strategy. The work provides concrete examples (e.g., in Set, Top, Vect) and discusses centralisable versus non-centralisable cases, along with central graded submonads and lax commutativity. Future directions include generalizing to pomonoid gradings, exploring different grading choices and inter-category morphisms, and linking these structures to graded computational effects and coeffects.
Abstract
We introduce the notion of 'centre' for pomonoid-graded strong monads which generalizes some previous work that describes the centre of (not graded) strong monads. We show that, whenever the centre exists, this determines a pomonoid-graded commutative submonad of the original one. We also discuss how this relates to duoidally-graded strong monads.