Dr. Dobb's | Category Theory

Category Theory

Here is a dialogue I read recently about implementing Windows XP:

Daniel de França: Hi John! If I am very patient, may I be able to write Windows XP using manifolds and cobordisms? […]
John Baez: If string theory is correct, Windows XP already runs using manifolds and cobordisms. If loop quantum gravity is correct, it’s using something a bit different, but very similar: spin networks and spin foams.
If Microsoft’s Project Q succeeds, and you’re very patient, you may someday use a version of Windows XP that runs using a modular tensor category implemented via the fractional quantum Hall effect. Read the conclusions of our paper!

Manifold and cobordism are things used in topology: manifolds represent spaces, and cobordisms represent mappings between them. String theory is based on them, so if string theory is correct, XP already runs using manifolds and cobordisms. Like the rest of the Universe. But why should anyone discuss this? That dialogue took place on the n-Category Café blog, and was about a new paper by John Baez and Mike Stay called Physics, topology, logic and computation: a Rosetta Stone.. The paper explains how, using a branch of maths called category theory, one can show an equivalence between four sets of ideas from different regions of science. These are: manifolds and cobordisms, in topology; states and processes that transform them, in physics; datatypes and functions that map between them, in computing; propositions and proofs that prove one from another, in logic. This equivalence will help us design quantum computers. Which is what Microsoft are doing in Project Q! Category theory has (I believe) vast power and potential in other parts of computing too, and in cognitive science. So, with applications that include neural networks, quantum teleportation, spreadsheets, analogical reasoning and metaphor, software specification, the way humans recognise similar concepts, and General Systems Theory, I thought I'd blog a series of posts about uses of category theory.

July 23, 2008
URL:http://www.drdobbs.com/architecture-and-design/category-theory/228701515