Universal algebra and equational logic stand as well-established tools for reasoning about programs and program equivalences. Universal quantitative algebra and quantitative equational logic were introduced in 2016 as a natural extension of these to reason about program distances. I will present the result of a few years of work with Matteo Mio and Valeria Vignudelli on improving and streamlining the original paper.
A central result in the classical realm is the correspondence between algebraic theories and (finitary) monads on the category Set. While a complete axiomatization of monads corresponding to quantitative algebraic theories remains out of reach, I will show every lifting of a monad on Set with an algebraic presentation can be presented by a quantitative algebraic theory. This result encompasses all currently known applications.