Blog » Intergalactic Telefunctors and Quantum Entanglement
Posted on 1234713995|%A: %d %B, %Y|agohover
In a previous article I argued against the pernicious Container Fiction. Some respondents have quite reasonably responded that this seems a bit pedantic; it may be true that e.g. a list is not really a container, but it's such a useful metaphor that only a pedant would insist on banning it.
Indeed it is a very useful metaphor, but the motivation for exposing it as a "pernicious" myth is not linguistic fascism, but rather pedagogical clarity. The Container Fiction is only pernicious insofar as it interferes with a clear understanding of the underlying abstraction it attempts to capture: the functor as a special kind of relation that enables communication between two different and mutually incomprehensible universes. This is bound to happen, since the container fiction is deeply embedded in the world of set theory, and the key to understanding Haskell is category theory, which is a whole nother thing.
The list is probably not the best example, since it involves a lot of complexity, what with the free monad and so forth. A simpler example would be a pair (a,b). This can obviously considered a kind of container, but as metaphors go a somewhat less pernicious image is the chemical compound. A chemical compound has different properties than its consituents; carbon monoxide (CO) is quite a different thing than carbon and oxygen. It's not a container of its constituents, it's their compound. The relation of (the denoted value) of "(a,b)" to its "components" a and b is similar. It's a mereological relation, not a containership relation.
But even that obscures the true nature of the functor (). The value denoted by "(a,b)" is not a compound, it is a primitive, whole, opaque value. (Disregard for the sake of argument whether it is dependent on the human mind.) The compound is the expression "(a,b)" itself.
My proposition is that a category theoretic explanation can be clearer and simpler than such metaphorical expositions, leading to a solid articulated intuition of the structural nature of functors without resorting to colorful metaphors. Well, maybe one, but it's arguably not a metaphor. A full exposition would take some very careful writing and a bunch of diagrams, but here's a sketch of how it would work.
Start with a, b, and (). Symbols 'a' and 'b' name things in a home universe; '()' is (the name of) an intergalactic "telefunctor" that relates the home universe to an alien universe of pair values. "(a,b)" takes two primitive values from home to a third primitive value in the alien universe. It doesn't convey a and b to the alien universe; rather, it entangles them. IOW, "(a,b)" is a kind of name or definite description of a value that is just as primitive and indivisible as the value denoted by "a" but that lives in an alien universe. We could say that "(a,b)" represents the quantum entanglement of a and b with the value denoted by "(a,b)". Call that value x. It lives in an alien universe; we know nothing about its internal structure, nor what it looks like in its universe, nor how it relates to other elements of its universe. But we do know something about how it relates to values in our home universe. We obviously know it is related - via () - to a and b together. But we also know, because of the way we define () - since this is just a brief sketch I omit this definition - we know that x is related to a and b individually. Our definition of () goes with two other definitions: fst (a,b) = a, and snd (a,b) = b.
Now we have defined the value of () purely in terms of how it relates things across intergalactic (ok, couldn't resist a metaphor) boundaries - morphisms. We could go further and show how a and b can also be defined as morphisms - after all, they're just names denoting values, and those values are just as opaque and unapproachable as the value of "(a,b)". Our knowledge of how all these things behave is expressed in terms not of what they are, but in how they relate. You can judge a man by his friends; you can understand mathematical values by their relations to other values.
A diagram might represent values as blackened circles. I believe this approach leads to a very clear, simple, and reasonably complete exposition. The metaphor of alien universes and intergalactic telefunctors could of course be dispensed with, but it does add color without introducing semantic distortion.
The real virtue of this approach becomes more clear when we complement the above exposition of the object component of a functor with a similar account of the morphism component that takes functions from one realm to functions in the alien realm. The pedagogical payoff is the realization that a functor allows us to use what we know about values in one realm to "manipulate" values in another realm. The best metaphor for the functor is neither containment nor mereology, but quantum entanglement.
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