The pullback theorem for relative monads
Nathanael Arkor
Tallinn University of Technology
Abstract:
It is well known that, for a monad T, the category of free T-algebras (a.k.a. the Kleisli category) and the category of all T-algebras (a.k.a. the Eilenberg–Moore category) satisfy individual universal properties. What is less well known is that the two categories furthermore satisfy a joint universal property that relates the two: this is known as the pullback theorem for monads, and establishes that the category of T-algebras may be constructed as a certain pullback involve the category of free T-algebras. I shall describe the generalisation of this theorem to relative monads, and explain the reason the theorem holds. The key insight stems from a tight relationship between relative monads and loose-monads (a.k.a. promonads or arrows). I shall end by describing some applications of the theorem to categorical algebra.
This talk is based on forthcoming joint work with Dylan McDermott.