Tarmo Uustalu: 'Grading monads, comonads and distributive laws'

We are accustomed to using strong monads to organize effectful computations in programming language semantics and monoidal comonads in a similar vein to organize coeffectful computations. I will show how a quantitative aspect can be added by grading, leading to mathematization of the old idea of effect systems from program analysis. I will first introduce graded (strong) monads, outlining both their theory and programming language applications, and will then discuss graded monads of monoids, graded distributive laws of a monad over a monad, graded (monoidal) comonads and distributive laws of a comonad over a monad.
This is based on recent works by Soichiro Fujii, Shin-ya Katsumata, Paul-André Melliès, Flavien Breuvart, Marco Gaboardi, Dominic Orchard and myself.