The Delannoy promonoidal category for scheduling processes

Abstract

We’ll consider a compositionality issue that arises when considering multiscale organizations, ones in which a process can call other ones that act at a faster time-scale. In particular, I’ll explain an oversight that occurred in my most recent paper with Sophie, where we forgot to ask ourselves whether the multiplication operation of the free monad monad is monoidal, and it’s not; this led us to the false claim that its Kleisli category is monoidal. I realized that if I’d forgotten to check this key property then, carelessness aside, it suggests that I was missing intuition about what could go wrong there, and that it behooved me to understand the issues better. The problem, which basically involves scheduling processes, led me to make some numerical calculations that in turn led me to the Delannoy numbers. I will discuss a promonoidal category corresponding to these numbers, and what it has to do with scheduling processes. This is all work in progress (including some great discussions with Harrison Grodin), and I hope to make even more sense of it with the seminar participants!