I'm not trolling, we're just talking around one another. There are a significant number of concepts that aren't easily explained in plain english required to get at the heart of "what is a monad?". Even that question itself is already making an interesting choice of definition and set of questions that can be confusing unless analyzed carefully (in particular "Monad" as a noun can mean something very different from monad as an adjective and both meanings are frequently used in descriptions of "what monads are").

I'm attempting to be strongly consistent in everything I'm saying, but it's not necessarily possible to both do that and keep speaking simply.

The why of monads is that they're a very nice API or Interface that allows you to write easily analyzed and easily refactored programs. Most of the other descriptions are the why of a particular type which happens to instantiate the API.

'There are a significant number of concepts that aren't easily explained in plain english required to get at the heart of "what is a monad?".'

Indeed, it's arguably not even the right question. "When is something monadic?" is a better one; "monad" is an adjective, not a noun. In Java terms, "Monad" is an interface and something is "monadic" if it implements the interface. (Except Java's type system can not express "Monad", so that doesn't work as a concrete example, but the idea is there.)

I'm sure you've seen me claim this on /r/haskell, where I think people operate in a context where they find this so obvious that the very idea that there could be a problem is confusing... this is the sort of context that leads me to that claim, though. Regardless of whether it's a noun in math, in programming it's an adjective.

That sounds nice and all, but i still don't know why you contradicted yourself.

Okay, I'll explain it. Apologies for the apparent contradiction—I'm not trying to move the goal posts, just trying to avoid going into the weeds.

As I said, Python decorators are essentially just a syntax for function composition.

    @deco
    def g(args):           g := deco(g(args): body)
      body

where (g := a) means a is assigned to the name g. As it turns out, functions form a type which is a functor over the result type. The fmap (arrow map) is just function composition. In Haskell you would write this as

    instance Functor ((->) r) where
      fmap g f = g . f

where the (.) syntax is also function composition, very similar to the @deco syntax in Python. In this sense, I stated that @deco is a Functor.

Now, the function data type also admits a Monad instance (which some people abuse terminology to say "functions are monads" but I highly dislike that terminology when speaking carefully).

    instance Monad ((->) r) where
      return a = \_ -> a
      (f >>= g) x = g (f x) x

So in this way, the function type instantiates both Functor and Monad, but composition (which is (.) in Haskell and @deco is Python) is not directly related to the Monad.

Instead, a function's Monad instance allows us to "defer application" and translate something like

    \x -> (1 + x, 2 + x)

into

    do a <- (1+)
       b <- (2+)
       return (a, b)

In this format it's pretty stupid looking, but we also tend to call this monad instance the "Reader Monad" which indicates computations which occur under a fixed set of configuration information—some static global variables.

For instance, if I had a Key type which represented some encryption key I could change my encrypt and decrypt functions from

    encrypt :: Key -> String -> String
    decrypt :: Key -> String -> String

to

    encrypt :: String -> Reader Key String
    decrypt :: String -> Reader Key String

which, allows us to defer that extra `Key` argument until later.

So, that's what I meant by saying that @deco isn't the Monad that its Functor represents—it only has the composition aspect, not the deferred application aspect.