Is there any intuition to understand join two functions in Monad?

Question

join is defined along with bind to flatten the combined data structure into single structure.

From type system view, (+) 7 :: Num a =>

Answer 1

Going along with the traditional analogy of a monad as a context for computation, join is a method of combining contexts. Let's start with your example. join (+) 7. Using a function as a monad implies the reader monad. (+ 1) is a reader monad which takes the environment and adds one to it. Thus, (+) would be a reader monad within a reader monad. The outer reader monad takes the environment n and returns a reader of the form (n +), which will take a new environment. join simply combines the two environments so that you provide it once and it applies the given parameter twice. join (+) === \x -> (+) x x.

Now, more in general, let's look at some other examples. The Maybe monad represents potential failure. A value of Nothing is a failed computation, whereas a Just x is a success. A Maybe within a Maybe is a computation that could fail twice. A value of Just (Just x) is obviously a success, so joining that produces Just x. A Nothing or a Just Nothing indicates failure at some point, so joining the possible failure should indicate that the computation failed, i.e. Nothing.

A similar analogy can be made for the list monad, for which join is merely concat, the writer monad, which uses the monoidal operator <> to combine the output values in question, or any other monad.

join is a fundamental property of monads and is the operation that makes it significantly stronger than a functor or an applicative functor. Functors can be mapped over, applicatives can be sequences, monads can be combined. Categorically, a monad is often defined as join and return. It just so happens that in Haskell we find it more convenient to define it in terms of return, (>>=), and fmap, but the two definitions have been proven synonymous.