Monad in non-programming terms [duplicate]

Answer 1

In non-programming terms:

If F and G are a pair of adjoint functors, with F left adjoint to G, then the composition G.F is a monad.

Answer 2

Yes, Monads comes from a concept outside of haskell. Haskell have many terms and Ideas that have been borrowed from Category theory. This is one of them. So if this person who is not a programmer turns out to be a mathematician who have studied Category theory, just say: "a Monad is a monoid in the category of endofunctors."

Answer 3

I like to think of them as abstractions of computations that can be "bound." Or, burritos!

Answer 4

Is there some concept/thing outside of programming (outside of all programming, not just FP) which could be said to act or be monad-like in a significant way?

Yes, in fact there is. Monads are quite directly related to "possibility" in modal logic by an extension of the Curry-Howard isomorphism. (See: A Judgmental Reconstruction of Modal Logic.)

This is quite a strong relationship, and to me the concepts related to possibility on the logical side are more intuitive than those related to monads from category theory. The best way I've found to explain monads to my students draws on this relationship but without explicitly showing the isomorphism.

The basic idea is that without monads, all expressions exist in the same world, and all calculation is done in that world. But with monads there can be many worlds and the calculation moves between them. (e.g., each world might specify the current value of some mutable state)

In this view, a monad p means "in a possible reachable world from the current world".

In particular if t is a type then:

x :: t means something of type t is directly available in the current world
y :: p t means something of type t is available in a world reachable from the current one

Then, return allows us to use the current world as a reachable one.

return :: t -> p t

And >>= allows us to make use of a something in a reachable world and then to reach additional worlds from that world.

(>>=) :: p t -> (t -> p s) -> p s

So >>= can be used to construct a path to a reachable world from smaller paths through other worlds.

With the worlds being something like states this is pretty easy to explain. For something like an IO monad, it's also pretty easy: a world is specified by all the interactions a program has had with the outside world.

For non-termination two worlds suffice - the ordinary one, and one that is infinitely far in the future. (Applying >>= with the second world is allowed, but you're unlikely to observe what happens in that world.) For a continuation monad, the world remains the same when continuations are used normally, and there are extra worlds for when they are not (e.g., for callcc).