Apply ?= Applicative, Chain ?= Monad

I’m a little confused here.

Is there any Functor that doesn’t have a .of function? Isn’t that just a constructor?

Just.prototype.of = function(x) {
  return new Just(x)
}

By this logic, an Apply is an Applicative and a Chain is a Monad.

Are there any good examples of data structures that are particularly only some of these categories, but not all of them?

[puffnfresh]

I just remembered that there’s a blog post with some examples, too: http://blog.functorial.com/posts/2015-12-06-Counterexamples.html

[ccorcos]

Where would it get the string? That’s the fundamental problem.

I see. So .of can take only a single parameter. And that parameter must be a basic type? Well what is a basic type? For example, what if you passed an array of two items in Pair.of(['foo', 7])?

What I’m really struggling with is discriminative examples. I finally understand how all this stuff works in regards to Maybes and Arrays, but thats because they work for just about all of them. But what are some examples that discriminate between these categories well?

• whats a semigroup that isn’t also a monoid?

• whats a foldable that isn’t also a traversable?

• whats an functor that isn’t also an apply?

• whats an apply that isn’t also an applicative?

  A pair is an Apply (you can bias map and ap to the right value), but it is not an Applicative because you can only pass one value (and any value) into .of and even if its just the right value, it will not pass the interchange law.

• whats an apply that isn’t also a chain?

• whats a chain that isn’t also a monad?

• what an applicative that isn’t also a monad?

I’ve been trying to find answers to these all damn week! Any ideas?