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Tue, 20 Jan 2009

Triples and Closure
Lately I've been reading Lambek and Scott's Introduction to Higher-Order Categorical Logic, which is too advanced for me. (Yoneda Lemma on page 10. Whew!) But you can get some value out of books that are too hard if you pay attention. Last night I learned that monads are analogous to closure operators.

In topology, we have the idea of a "closure" of a set, which is essentially the union of the set with its boundary. For example, consider an open disk D, say the set of all points less than one mile from my house. The boundary of this set is a circle with radius one mile, centered at my house. The closure of D is the union of D with its boundary, and so is closed disk consisting of all points less than or equal to one mile from my house.

Representing the closure of a set S as C(S), we have the obvious theorem that SC(S), because the closure includes everything in S, plus the boundary.

Another easy, but not quite obvious theorem is that C(C(S)) ⊂ C(S). This says that once you take the closure, you have included the boundary, and you do not get any more boundary by taking the closure again. The closure of a set is "closed"; the closure of a "closed" set C is just C.

A third fundamental theorem about closures is that ABC(A) ⊂ C(B).

Now we turn to monads. A monad is first of all a functor, which, if you restrict your attention to programming languages, means that a monad is a type constructor M with an associated function fmap such that for any function f of type α → β, fmap f has type M α → M β.

But a monad is also equipped with two other functions. There is a return function, which has type α → M α, and a join function, which has type M M α → M α.

Haskell provides monads with a "bind" function, written >>=, which is interdefinable with join:

        join x   =  x >>= id
        a >>= b  =  join (fmap b a)
but we are going to forget about >>= for now.

So the monad is equipped with three fundamental operations:

        fmap   ::   (a → b) → (M a → M b)
        join   ::   M M a → M a
        return ::   a → M a
The three basic theorems about topological closures are:

(AB) → (C(A) ⊂ C(B)
C(C(A)) ⊂ C(A)
AC(A)
If we imagine that ⊂ is a special kind of implication, the similarity with the monad laws is clear. And ⊂ is a special kind of implication, since (AB) is just an abbreviation for (xAxB).

If we name the three closure theorems "fmap", "join", and "return", we might guess that "bind" also turns out to be a theorem. And it is, because >>= has the type M a → (aM b) → M b. The corresponding theorem is:

xC(A) → (AC(B)) → xC(B)
If the truth of this is hard to see, it is partly because the implications are in an unnatural order. The theorem is stated in the form PQR, but it would be easier to understand as the equivalent QPR:

AC(B) → xC(A) → xC(B)
Or more briefly:

AC(B) → C(A) ⊂ C(B)
This is quite true. We can prove it from the other three theorems as follows. Suppose AC(B). Then by "fmap", C(A) ⊂ C(C(B)). By "join", C(C(B)) ⊂ C(B). By transitivity of ⊂, C(A) ⊂ C(B). This is what we wanted.

Haskell defines a =<< operator which is the same as >>= except with the arguments forwards instead of backwards:

        =<< ::  (a → M b) → M a → M b
        a =<< b   =   b >>= a
The type of this function is analogous to the bind theorem, and I have seen claims in the literature that the argument order is in some ways more natural. Where the >>= function takes a value first, and then feeds it to a given function, the =<< function makes more sense as a curried function, taking a function of type aM b and yielding the corresponding function of type M aM b.

I think it's also worth noticing that the structure of the proof of the bind theorem (invoke "fmap" and then "join") is exactly the same as the structure of the code that defines "bind".

We can go the other way also, and prove the "join" theorem from the "bind" theorem. The definition of join in terms of >>= is:

        join a = a >>= id
Following the program again, id in the program code corresponds to the theorem that BB for any B. A special case of this theorem is that C(B) ⊂ C(B) for any B. Then in the "bind" theorem:
AC(B) → C(A) ⊂ C(B)
take A = C(B):

C(B) ⊂ C(B) → C(C(B)) ⊂ C(B)
The left side of the implication is satisfied, so we conclude the consequent, C(C(B)) ⊂ C(B), which is what we wanted.

But wait, monad operations are also required to satisfy some monad laws. For example, join (return x) = x. How does this work out in topological closure world?

In programming language world, x here is required to have monad type. Monad types correspond to closed sets, so this is a theorem about closed sets. The theorem says that if X is a closed set, then the closure of X is the same as x. This is true.

The identity between these two things can be found in (surprise) category theory. In category theory, a monad is a (categorial) functor equipped with two natural transformations, the "return" and "join" operations. The categorial version of a closure operator is essentially the same.

Closure operations have a natural opposite. In topology, it is the "interior of" operation. The interior of a set is what you get if you discard the boundary of the set. The interior of a closed disc is an open disc; the interior of an open disc is the same open disc. Interior operations satisfy laws analogous but opposite to those enjoyed by closures:

SC(S) I(T) ⊂ T
C(C(S)) ⊂ C(S) I(T) ⊂ I(I(T))
ABC(A) ⊂ C(B) ABI(A) ⊂ I(B)
Notice that the third theorem does not get turned around. I think this is because it comes from the functor itself, which goes the same way, not from the natural transformations, which go the other way. But I have not finished thinking abhout it carefully yet.

Sooner or later I am going to program in Haskell with comonads, and it gives me a comfortable feeling to know that I am pre-equipped with a way to understand them as interior operations.

I have an idea that the power of mathematics comes principally from the places where it succeeds in understanding two different things as aspects of the same thing. For example, why is group theory so useful? Because it understands transformations of objects (say, rotations of a polyhedron) and algebraic operations as essentially the same thing. If you have a hard problem about one, you can often make it into an easier problem about the other one. Similarly analytic geometry transforms numerical problems into geometric problems and back again. Most often the geometry is harder than the numerical problem, and you use it in that direction, but often you go in the other direction instead.

It is quite possible that this notion is too vague to qualify as an actual theory. But category theory fits the description. Category theory lets you say that types are objects, type constructors are functors, and polymorphic functions are natural transformations. Then you can understand natural transformations as structure-preserving maps of something or other and get some insight into polymorphic functions, or vice-versa.

Category theory is a large agglomeration of such identities. Lambek and Scott's book starts with several slogans about category theory. One of these is that many objects of interest to mathematicians form categories, such as the category of sets. Another is that many objects of interest to mathematicians are categories. (For example, each set is a discrete category.) So one of the reasons category theory is so extremely useful is that it sets up these multiple entities as different aspects of the same thing.

I went to lunch and found more to say on the subject, but it will have to wait until another time.


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