Pairs, sums, and reactivity
I’ve been noodling over a difference between behaviors (functions of time) and events (sequences of time/value pairs). A product of behaviors is isomorphic to a behavior of products, but a product of events is isomorphic to an event of sums.
Ba × Bb ≅ Ba × b
Ea × Eb ≅ Ea + b
A similar property has been noted for Fudgets-like stream processors.
Behaviors (or "reactive values") and events are inter-related. In particular, we can make a behavior from an initial value and an event, using the stepper function. If we want to use stepper with a pair value, we’d have to use a pair-valued event. However, it’s often more convenient and efficient to work with pair of separate change events, or (using the event pair/sum isomorphism) a sum-valued event.
This post plays with another perspective on sum types for events and stream processors. It can also apply to bots.