[MaxRegret]

I think the article is an attempt to make your idea more precise. What if monad 3 could use the result of monad 1, not just monad 2? Then you couldn't write that computation using your pipeline metaphor.

The article is saying that by judicious use of a "strength" function, you can in fact write any monadic computation as such a pipeline (or composition of monadic functions).

[koolala]

If my example of pipeline is a limited metaphor, couldn't it be expanded?

I love this subject but it's traditionally over-complex, so I appreciate the correction and explaination.

How could "strength" be applied generically to programs outside of Haskell? Could the concept of pipelines be extended? Is programming too 1 dimensional? There must be ways to do this.

[MaxRegret]

I think the user-friendly way to make programming less one-dimensional is what we usually do, outside of shell pipelines; give names to intermediate values:

  do x <- computation1
     y <- computation2 x
     computation3 x y

...turning a one-dimensional sequence of operations into a data-flow graph represented textually.

But, and perhaps this is an academic question, what if you're stuck in a context where you can only create computations by composing functions in a sequence, without giving names to intermediate results? Shell pipelines, for example, without using temporary files or other tricks.

Then, using this idea of strong and weak monads, and if you were used to thinking of computations as elements of a monad type, you might realize that all you need to add is a "strength" operator, and you would regain the full expressivity of data flow graphs. This operator would take a computation and bundle it with value, producing a new computation that does the same computation, but bundles its result with the extra value you supplied. So you've given every computation a side-channel that ferries any data you like through it, unchanged. Computations further down the chain can access both values.

I lack the imagination to see how this example works out in shell, because there we don't have the luxury of multiple input and output values (just single pipes of data). But it's nice to know that just the strength operation is enough to express arbitrary computation graphs, with names being given to inputs and outputs.