# reperiendi

Posted in Category theory, Math, Programming by Mike Stay on 2012 August 28

With the new ECMAScript 6 Proxy object that Firefox has implemented, you can make dot do pretty much anything you want. I made the dot operator in JavaScript behave like Haskell’s bind:

```// I'll give a link to the code for lift() later,
// but one thing it does is wrap its input in brackets.
lift(6); // [6]
lift(6)[0]; // 6
lift(6).length; // 1

// lift(6) has no "upto" property
lift(6).upto; // undefined

// But when I define this global function, ...

// Takes an int n, returns an array of ints [0, ..., n-1].
var upto = function (x) {
var r = [], i;
for (i = 0; i < x; ++i) {
r.push(i);
}
return r;
};

// ... now the object lift(6) suddenly has this property
lift(6).upto; // [0,1,2,3,4,5]
// and it automagically maps and flattens!
lift(6).upto.upto; // [0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]
lift(6).upto.upto.length; // 15
```

To be clear, ECMAScript 6 has changed the API for Proxy since Firefox adopted it, but you can implement the new one on top of the old one. Tom van Cutsem has code for that.

I figured this out while working on a contracts library for JavaScript. Using the standard monadic style (e.g. jQuery), I wrote an implementation that doesn’t use proxies; it looked like this:

```lift(6)._(undo)._(undo).value; // [0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]
```

The `lift` function takes an input, wraps it in brackets, and stores it in the `value` property of an object. The other property of the object, the underscore method, takes a function as input, maps that over the current value and flattens it, then returns a new object of the same kind with that flattened array as the new value.

The direct proxy API lets us create a “handler” for a target object. The handler contains optional functions to call for all the different things you can do with an object: get or set properties, enumerate keys, freeze the object, and more. If the target is a function, we can also trap when it’s used as a constructor (i.e. `new F()`) or when it’s invoked.

In the proxy-based implementation, rather than create a wrapper object and set the `value` property to the target, I created a handler that intercepted only get requests for the target’s properties. If the target has the property already, it returns that; you can see in the example that the length property still works and you can look up individual elements of the array. If the target lacks the property, the handler looks up that property on the window object and does the appropriate map-and-flatten logic.

I’ve explained this in terms of the list monad, but it’s completely general. In the code below, `mon` is a monad object defined in the category theorist’s style, a monoid object in an endofunctor category, with multiplication and unit. On line 2, it asks for a type to specialize to. On line 9, it maps the named function over the current state, then applies the monad multiplication. On line 15, it applies the monad unit.

```var kleisliProxy = function (mon) {
return function (t) {
var mont = mon(t);
function M(mx) {
return Proxy(mx, {
get: function (target, name, receiver) {
if (!(name in mx)) {
if (!(name in window)) { return undefined; }
return M(mont['*'](mon(window[name]).t(mx)));
}
return mx[name];
}
});
}
return function (x) { return M(mont[1](x)); };
};
};

var lift = kleisliProxy(listMon)(int32);
lift(6).upto.upto; // === [0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]
```