Monad is a way to preserve symmetries in the space of functions while allowing for maximal compositionality. It is also Leibniz's way^[1][2] to stating the notion of Digital Twin and Metaverse.

Monad: Natural numbers as Functors

To model numbers in terms of relations, monad can be used as a bridge. That is based on the fact that functors can be used to represent both elements in a set and the relations of the elements in the set. In other words, the notion of representable is inalienable from the notion of functor, which carries the name of this information compression. Daniel Tubbenhauer's VisualMath also has a video on What are…monads?^[3]. In the beginningof the video, he stated that monad is a way of counting.

Don't fear the Monad

Brian Beckman had an hour-long video that explains Monad in a rather detailed and insightful tutorial^[4]. Therefore, a special page was dedicated to annotate the tutorial with segmented video clips. Just click on this link:Don't fear the Monad, to get access to the annotation. The following is the shortened version of the page:

Synopsis of Beckman's Tutorial on Monad

1. Monad is the way to build complexity from simplicity
2. Monad is ruled by one customizable rule that rules them all
3. Monad is hard to learn because of a broken symmetry

The idea of Monad can be traced back to the mathematical structure: Monoid, which is just an algebra with one element only. This mathematical structure provides a basis to reduce complexity. As Beckman says:

Monoid helps to guarantee you to build a software with one and only one type ...(start at 1126)

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