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...

Sunday, February 19, 2017

Monoids: the root of it all

Let's start talking about category theory. We will start from set theory and in the end try to get away from it. The first thing we need to discuss is magma. Basically you have a binary operation on a set and that's all: \(M \times M\rightarrow M\). One problem with magmas is that there is no associativity. Now not all mathematical operations lacking associativity are inherently primitive. Think of Lie algebras: the operation is not associative. However there you have something else: the Jacobi identity. But a pure magma without any additional structure is a rather inert object. The other problem with magmas is the lack of a unit. Add associativity and a unital element and category theory comes alive. 

To link the discussion to physics, nature obeys the structure of a (commutative) monoid: two physical systems can be composed into a larger physical system:
- composition is the binary operation 
- associativity guarantees our ability to reason about physical systems regardless of how we split a physical system into subsystems: quantum mechanics is valid for both an electron or an atom containing an electron
- the unital element is nothingness: composition with nothing leaves the original physical system intact.

In later posts I will show how quantum mechanics is a logical consequence of the commutative monoid above. In other words, quantum mechanics is inescapable and nature is quantum all the way.

Back on monoids, let's fall back on the usual example: composable functions: the image of a function is the domain of the next function. The link with programming is obvious: the output of one computation is plugged in as the input of another computation. As a side note, because of this functional programming is best explained in the language of category theory. When we talk function composition we usually write: \(f \circ g\)  which means \(f(g(x))\). To jump in abstraction and eliminate the nature of the elements considerations, there is an elementary trick to help navigate complex composition chains: call \(\circ\): AFTER like this: \(f~composed~with~g = f\circ g = f~ AFTER~ g\)

Now let's review the usual properties of injectivity and surjectivity:

Injectivity: for any elements \(x, x^{'}\), a function is injective if \(f(x) = f(x^{'}) ~implies~ x=x^{'}\)
Surjectivity: for any \(y\) in the range, there is an \(x\) in the domain such that \(f(x)=y\)

So how can we abstract this away and eliminate the talk about the elements? The corresponding category theory concepts are monic and epic:

Monic: a morphism is monic if for any \(g, h\) \(f\circ g = f\circ h ~implies~g=h\)
Epic: a morphism is epic if for any \(g, h\) \(g\circ f = h\circ f ~implies~g=h\)

Can you prove that if \(f:X\rightarrow Y\) then \(f\) is injective if and only if it is monic and it surjective if and only if it is epic? The proof can be found in many places but it is instructive to try to prove it yourself without looking it up first as this will help you better understand category theory. 

The last point I want to make today is that in category theory we move away from functions into abstract morphisms. The key point of morphisms is that they preserve mathematical structures. As such they can be used to jump between categories of very different nature. This is how category theory is a unifying structure of mathematics where the same patterns of reasoning can be replicated from logic to computer science, to algebraic topology, to quantum mechanics.

To be continued...

Sunday, February 12, 2017

A new way to look at mathematics

I want to start today  a series of posts about category theory. This is a vast area of mathematics which unifies logic, computer programming, combinatorics, cohomology, etc, and quantum mechanics into a cohesive paradigm. It also settles the problem of interpretation for quantum mechanics. By its very construction category theory has no need for any realism baggage. The entire mathematics can be expressed not in the language of sets (which are abstractions based on our classical intuition) but in the language of categories free of any considerations about the nature of elements. Regarding physics, the paradigm of category theory is best expressed by a famous Bohr quote:

"It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we say about Nature."

Let me start slow. The usual usage of math is on the practical side to solve problems. How many times did we hear the lazy student complaint: why should we learn this? Math is not about memorization and math is very easy once we absorb its content. Learning math is a journey in mastering abstractions and general ways of reasoning. For example when you learn about Lie groups you can extract a lot of key result by elementary methods simply by studying matrices. However you hit a wall with octonions because they are not associative and do not admit a matrix representation for this very reason. In turn this precludes the proper understanding of exceptional Lie groups.

Or consider a simpler example, topology. A lot of functional analysis can be done using the concept of distance and metric spaces. For example a space in \(R^n\) is compact iff it is closed and bounded. Then the metric spaces are generalized by the concept of topological spaces which are based on the idea of neighborhoods, unions, and intersections. In this case compactness is defined much more abstractly: a space is compact iff any open covering has a finite subcover. 

A similar thing happens in category theory. Patterns of reasoning in various mathematical domains are abstracted away in a formalism which does not care about the nature of the elements. On one end this is harder and to help navigate this in the beginning you hold on particular examples; the typical examples are functions. However at some point you let go of the examples just like in topology you let go the notion of distance. At that point you learn to reason properly in category theory and a lot can be achieved in this way. Then we can make the journey backwards from abstract to concrete. There is a big bonus in this: we have the flexibility to pick the concrete examples we want. And in our case we will pick quantum mechanics. Quantum mechanics is best and most naturally expressed in the language of category theory. Goodbye sets, goodbye classical realism, let the category journey begin. Please stay tuned.