`join` removes one layer of a monadic context: given `F[F[A]]`, it produces `F[A]`. It is the flattening operation behind a useful decomposition of `flatMap`: first `map` a function that returns `F[B]`, producing `F[F[B]]`; then `join` the result. section 11.7 section 7.4
def join[A](ppa: Par[Par[A]]): Par[A]
The type states the entire job: a nested parallel computation becomes one parallel computation. section 7.4
For `Par`, joining means running the outer computation, waiting for its result to reveal an inner computation, then running that inner computation. This lets later parallel work be selected by an earlier result rather than fixed in advance. section 7.9 section 12.3.1
`join` and the other monadic combinators can define one another:
- Define `join` with `flatMap(identity)`: the contained value is already the next monadic computation. section 7.9 section 11.7
- Define `flatMap(f)` as `map(f).join`, separating nesting from flattening. section 7.9 section 11.7
- Use `unit`, `map`, and `join` as a minimal basis for a monad, alongside equivalent bases using `flatMap` or `compose`. section 11.4.3 section 11.7
Flattening a parallel computation
If mapping `f: A => Par[B]` over `pa: Par[A]` yields `Par[Par[B]]`, `join(pa.map(f))` yields `Par[B]`. In the direct implementation, the outer `Par` is run and awaited; its returned `Par` is then run on the same executor service. section 7.9
Flattening is capability that `Applicative` operations do not supply. `unit` can add a context layer, and `map2` can combine values within contexts, but neither removes an `F` layer. Consequently, dependent computations require monadic operations such as `join` or `flatMap`. section 12.3 section 12.3.1
The associativity law for `join` says that a three-layer value can be flattened from the outside or inside first with an equivalent result: `x.join.join == x.map(_.join).join`. For `Par`, this equates alternative orders of awaiting nested computations; for parsers, it preserves the order of dependent parsers regardless of nesting. section 11.7
