Tuesday, February 28, 2017

Objects and arrows

With one day delay, let's continue the discussion about category theory. One way to look at category theory is as a generalization of the notion of equivalence: category theory = equivalence on steroids

It is informative to look at the original motivation for category theory and also to look at a problem around 1900. Suppose you go back in time without knowing any modern math except group theory and you are aware of Mobius strip, and Klein bottle. Your task is to try to figure out what else is possible? In other words, classify all two dimensional surfaces. Who can help you on this quest? Well, clothes are two dimensional surfaces made by tailors. How do they make them? By two operations: cutting and stitching. Knowing group theory, you realize cutting and stitching are opposite operations, and they do respect the axioms of a group. This is how homology theory actually came from: associating groups with topological spaces in order to classify them. Now fast forward to 1940s, several homology theories were known and the problem was why the groups involved in them are the same? How do we axiomatize homology theory and how do we know if two homologies are equivalent? The answer lies in the concept of natural transformation which requires the concept of functor, which in turn needs the idea of a category. 

So what is a category? A category consists of objects and morphisms (arrows) such that the morphisms can be composed. Here are some examples:

-examples from math:
  • sets and functions
  • groups and group homomorphisms
  • Hilbert space and operators
  • partial order sets and monotone functions
  • manifolds and cobordisms
-examples from logic
  • propositions and proofs
-example from physics
  • physical systems and physical processes
-examples from computer science
  • data types and programs
Now a functor maps a category to another category by mapping objects to objects and arrows to arrows in a way that preserves structure. This is how for example in algebraic topology we associate groups to topological spaces. 

A natural transformation is a arrow (morphism) between functors subject to some (natural) conditions.  

Apart of naturality, another key concept is universality which means a unique (up to an isomorphism) solution to problems of constructions. We will encounter that when we will express quantum mechanics in category formalism.

Category theory reveals surprising relationships:
  • Cartesian  products of sets are like greater lower bounds of partial order sets
  • Proofs in logic are like programs in functional programming
Back to quantum mechanics, unitary evolution preserves information and it should be no surprise that there quantum information can be represented in diagramatic fashion. However this is not the path I am going to take and I will make use of universality in deriving quantum mechanics from a simple principle - composition: a theory T describing two physical systems A and B must described the composite system A+B as well. This is very intuitive principle but in the formalism of category theory it has extremely powerful mathematical consequences, spelling out the complete internal details of the theory T. Quantum mechanics comes out of this in its full detail. 

Please stay tuned...

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