From Graded Foundations to the Foundations of Grading

Abstract

In recent decades, monads have come to be an indispensable tool in programming language theory and practice. Graded monads, which generalise monads by allowing the operations of a monad to be ‘stratified’ by a monoid, are a relatively recent innovation which provides the same utility as monads but utilises the monoid structure to provide more fine-grained control. As the use of graded monads rises so does the need to understand them formally. This work examines graded monads as lax 2-functors, to build up to a 2-category of graded monads. This provides concrete definitions of morphisms between graded monads, graded monad distributive laws, and composition of graded monads. We then demonstrate the utility of the 2-category of graded monads by providing the free graded monad construction. Further we show how using the notion of internal graded monad, we can define graded multicategories–which can be used to model graded base logics–as an extension of Leinster’s generalized multicategories.
(note: seminar will be at 2pm!)