Saturday, February 7, 2015
Does jQuery expose a monadic interface?

Pam Selle raised an interesting question on her blog recently: does jQuery expose a monadic interface? I'll interpret this as specifically asking: are jQuery containers monads? This was such an interesting question that I decided to investigate it more thoroughly; one can never have too much practice figuring out how monads work, in my experience!

There are a couple of formulations of monads; both have in common that there is a container type M that "wraps" plain data values in some fashion. Here, M is the type of jQuery containers, and the plain data values are DOM elements. Some monads are polymorphic, in that they can wrap values of a wide variety of underlying data types (lists and Option/Maybe are good examples of these). jQuery containers are also polymorphic, although we usually just apply them to DOM elements. For the rest of this post, we'll refer to C as the type of a jQuery container, and when it contains items of type T then we'll write C[T].

Monadic operators

Now, in order to qualify as a monad, a type has to support certain operations with particular signatures, and then those operations have to obey certain "monadic laws" in order to be correctly composeable.

Both formulations share a requirement for a simple wrapping function called return or unit that take plain data values (DOM elements) and return values of the monadic type (jQuery containers, or C). In other words, we're looking for an operation / "constructor" of type:

unit : T → C[T]

If e is a DOM element, then $(e) returns a jQuery container with that one element in it; ditto for $(a) if a is an array of DOM elements. We can then understand all the other selectors as syntactic sugar for one or the other of these. For example, $("#foo") is equivalent to $(document.getElementById("foo")). In fact, we can see that even just $(elt) for a single element is equivalent to $([elt]) (i.e. wrapping the element in a single element array). This will be convenient for the rest of this post, since it means we just have to deal with this simplest constructor. At any rate, it seems like we have the unit operation covered.

Now, in one of the formal definitions of monad, the other operation required is one called bind that can take a value of the monadic type, a function from the underlying data type to another value of the monadic type, and returns a value of the monadic type again. That's a mouthful, so it might be helpful to look at the type of this operator:

bind : C[T] → (T  C[U]) → C[U]

Conceptually, this takes every contained item of type T, applies a transformation to it that maps to a new set of containers, and then "squashes" it down into one container again. As is common with object-oriented language implementations, the this variable can be thought of as an implicitly-passed parameter, so we can then look through the API for a jQuery container looking for a method that takes one of these transformation callbacks and returns a new jQuery container. One such candidate is the .map() method, which is defined as:
Pass each element in the current matched set through a function, producing a new jQuery object containing the return values.
Wow, this looks pretty good; it takes a transformation function and returns us a jQuery container at the end. The real question is whether it will "flatten" things down into a single jQuery container for us (since the documentation doesn't say) if our function returns jQuery containers. Suppose we have the following markup embedded in a page:
<div id="example">
  <div class="outer"><div id="one" class="inner"></div></div>
  <div class="outer"><div id="two" class="inner"></div></div>
Now, suppose we try this in the Javascript console:
> $(".outer").map(function (idx,elt) { return $(elt).children(); });
We can see that this does, in fact, squash down to just a container with the two div.inner elements in it, as opposed to a container containing two containers. Nice!

Monad Laws

We have our candidate operations for unit and bind, so we have to check whether they adhere to certain properties; these properties are similar in spirit to the commutative or associative properties of addition (or, perhaps better, the distributive law of multiplication over addition, since that expresses a relationship between two operations). For the below notation, the bind operator is written as >>=. The first law says:

(unit x) >>= f   ≅   f x

Or, to put things into jQuery syntax: $(x).map(f) should be the same as f(x). If f() can take a DOM element and return a jQuery container, then we can see from our test above that we're in good shape here. The second law says:

m >>= return   ≅   m

which is to say that if m is a jQuery container, then it should be the case that (idx,item) { return $(item); }) gives us m back again, which also seems right. Finally, the third law says:

(m >>= f) >>= g   ≅   m >>= ( \x→ (f x >>= g))

Or, that if m is a jQuery container, and f and g are transformer functions, that the following are equivalent: (idx,x) { return f(x).map(g); })

Ok, that's a mouthful--or at least a keyboardful. What we are doing in the first line is taking each element in m and applying f to it, squashing this into one container, then taking each element of that collection and then applying g to it, then flattening everything down into one colleection. In the second line, we are taking each element in m, and then applying the given function to it, but we can see that the body of the function does the same thing: namely, applying f to the element (which returns a container, remember), and then mapping g across all those elements. Because .map() does squashing for us, we end up with the equivalent containers at the end.

And that's it! It does look like jQuery containers are a monad after all. We can actually understand several methods of jQuery containers as convenient applications of .map(). For example, .children() is really equivalent to:

.map(function (idx,elt) {
       return $(elt.childNodes);

[I suspect that the actual implementation of .children() ends up being more efficient, as it doesn't have to construct all the intervening jQuery containers and then squash their arrays back together again.]

Exercise for the reader: Can you find an expression of jQuery containers as a monad in their other formulation, with return, fmap, and join? (I ran out of time to try this before I had to make dinner for my kids!)

The Upshot

"So what?" you might ask. Well, it suggests that if you are writing a library meant to be used with jQuery, you might be well served to write many of your utility functions in the form of the monadic transform functions we saw above, taking DOM elements and returning jQuery containers--i.e. making sure your utility functions can be passed as arguments to .map(), because it means that they will be composeable in very flexible ways that let them be chained together easily.