kleisli | Usable , idiomatic common monads in Ruby | Functional Programming library

Quick question

With

I know >>= determines the Kleisli composition operator >=> (as explained here), but does it ever arise in mathematics per se.

How can I compose two kleisli arrows(functions) f:A -> Promise B and g: B -> Promise C into h:A -> Promise C using fp-ts?

I’m familiar with Haskell so I would ask in this way: What’s the equivalent of >=>(fish operator)?

Some definition of Monad Transformers in cats.

EitherT[F[_], A, B] is a lightweight wrapper for F[Either[A, B]] that makes it easy to compose Eithers and Fs together. To use EitherT, values of Either, F, A, and B are first converted into EitherT, and the resulting EitherT values are then composed using combinators.

OptionT[F[_], A] is a light wrapper on an F[Option[A]]. Speaking technically, it is a monad transformer for Option, but you don’t need to know what that means for it to be useful. OptionT can be more convenient to work with than using F[Option[A]] directly.

I understand the concept, even saw some interesting talk about it: Monad transformers down to earth by Gabriele Petronella

I understand the Reader Monad, and how Kleisli is just a generalization.

What i do not understand is the statement bellow, that says that it is a monad transformer. What are we stacking exactly ? I don't see 2 monad being stacked here ....

Kleisli can be viewed as the monad transformer for functions. Recall that at its essence, Kleisli[F, A, B] is just a function A => F[B], with niceties to make working with the value we actually care about, the B, easy. Kleisli allows us to take the effects of functions and have them play nice with the effects of any other F[_].

Any thoughts ?

It is widely known that Applicative generalises Arrows. In the Idioms are oblivious, arrows are meticulous, monads are promiscuous paper by Sam Lindley, Philip Wadler and Jeremy Yallop it is said that Applicative is equivalent to Static Arrows, that is arrows for which the following isomorphism holds:

arr a b :<->: arr () (a -> b)

As far as I can understand, it could be illustrated the following way:

Note: newtype Identity a = Id { runId :: a }.

Klesli Identity is a Static Arrow as it wraps k :: a -> Identity b. Isomorphism just removes or adds the wrapper.

Kleilsi Maybe is not a Static Arrow as k = Kleisli (const Nothing) exists - all f :: a -> bs correspond to Just . f, and there is no place for k in the isomorphism.

But at the same time both Kleisli Identity and Kleisli Maybe are Arrow instances. Therefore, I can not see how the generalisation works.

In Haskell/Understanding Arrows tutorial on Wikibooks they say static morphism and note the following:

Those two concepts are usually known as static arrows and Kleisli arrows respectively. Since using the word "arrow" with two subtly different meanings would make this text horribly confusing, we opted for "morphism", which is a synonym for this alternative meaning.

That is the only lead I have so far - am I confusing Haskell Arrow and arrows?

So, how does this hierarchy work? How is this Applicative's property formalised/proven?

repmin problem is pretty well-known. We are given a data type for trees:

trying to implement Kleisli category for a made-up Partial type in Scala (reading Bartosz Milewski's "category theory for programmers", that's exersize for chapter 4)

I have a call chain of some methods, where I pass a context via a Kleisli. Basically I want to pass a context down to the db access layer, but I want to access this context everywhere in between.

The following example works perfectly. My problem though is, that I want to access the context in OrderService.findAll(...) as well. I tried several approaches but I keep failing.

I'm trying to write middleware that would extract specific cookie and store information in ContextRequest. Here is my test code: