Quantum theory can be thought of as the generalization of classical mechanics you get by dropping the assumption that observable quantities like position and momentum commute. In quantum theory one thus learns to like noncommutative, but still associative, algebras.
It is interesting however to note why associativity without commutativity is studied so much more than commutativity without associativity. Basically, because most of our examples of binary operations can be interpreted as composition of functions. For example, if write simply x for the operation of adding x to a real number (where x is a real number), then x + y is just x composed with y. Composition is always associative so the + operation is associative!
If we try to generalize the heck out of the concept of a group, keeping associativity as a sacred property, we get the notion of a category. Categories are some of the most basic structures in mathematics. They were created by Samuel Eilenberg and Saunders MacLane.
In fact, MacLane said: "I did not invent category theory to talk about functors. I invented it to talk about natural transformations." Huh? Wait and see.
First of all: what is a category? Well, a category consists of a set of objects and a set of morphisms. Every morphism has a source object and a target object. If f is a morphism with X as its source and Y as its target, we write
f: X → Y
and say f goes from X to Y. We write Hom(X,Y) for the set of morphisms from X to Y.
The example to keep in mind is the category in which the objects are sets and a morphism f: X → Y is a function from the set X to the set Y. Then Hom(X,Y) is the set of functions from X to Y.
The axioms for a category are that it consist of a set of objects and for any 2 objects X and Y a set Hom(X,Y) of morphisms from X to Y, such that:
The classic example is Set, the category with sets as objects and functions as morphisms, and the usual composition as composition! Or else:
Note that in all these cases the morphisms are actually a special sort of function. That need not be the case in general! For example, an ordered set is just a category with its elements as objects and one morphism in each Hom(X,Y) if X is less than or equal to Y, but none otherwise. Also, a group is just a category where there's one object and all the morphisms have inverses - we call the morphisms "elements" of the group. Weird, huh? It's actually a very useful viewpoint.
The golden rule of modern mathematics is that life takes place within - and between - categories. Many nice things in mathematics are functors. A functor is a kind of map between categories. A functor F from a category C to a category D is a map from the set of objects of C to the set of objects of D together with a map from the set Hom(X,Y) for any objects X,Y of C to Hom(F(X),F(Y)). That is, objects go to objects and morphisms go to morphisms. We demand that:
Category theory is popular among algebraic topologists. Typically an algebraic topologist will try to assign algebraic invariants to topological structures. The golden rule of such invariants is that they should be functorial. That is, they should be functors! For example, the fundamental group is functorial. Topologists know how to cook up a group called the fundamental group from any space. (The group keeps track of how many holes the space has.) But ALSO, any map between spaces determines a homomorphism of the fundamental groups. So the fundamental group is really a functor from the category Top to the category Group.
This allows us to completely transpose any situation involving spaces and continuous maps between them to a parallel situation involving groups and homomorphisms, and thus reduce some topology problems to algebra problems! For example, it's easy to use this idea to show that you can't map a disk continuously to its boundary in such a way that the points on the boundary get mapped to themselves. You take this problem in topology, turn it into an algebra problem, and presto! It's easy!
There is a famous saying about quantization due to Edward Nelson: "First quantization is a mystery, but second quantization is a functor!" No one is a true mathematical physicist unless they can explain that remark. So, let me explain that remark!
First quantization is a mystery. It is the attempt to get from a classical description of a physical system to a quantum description of the "same" system. Now it doesn't seem to be true that God created a classical universe on the first day and then quantized it on the second day. So it's unnatural to try to get from classical to quantum mechanics. Nonetheless we are inclined to do so since we understand classical mechanics better. So we'd like to find a way to start with a classical mechanics problem - that is, a phase space and a Hamiltonian function on it - and cook up a quantum mechanics problem - that is, a Hilbert space with a Hamiltonian operator on it. It has become clear that there is no utterly general systematic procedure for doing so.
Mathematically, if quantization were "natural" it would be a functor from the category whose objects are symplectic manifolds (= phase spaces) and whose morphisms are symplectic maps (= canonical transformations) to the category whose objects are Hilbert spaces and whose morphisms are unitary operators. Alas, there is no such nice functor. So quantization is always an ad hoc and problematic thing to attempt. A lot is known about it, but more isn't. That's why first quantization is a mystery.
(By the way, I have seen many "no-go" theorems concerning quantization but I have never seen one phrased quite like the above. "There is no functor from the symplectic category to the Hilbert category such that ..... holds." Is anyone up to the challenge?? If this hasn't been done yet it would clarify the situation.)
Note that there IS a functor from the symplectic category to the Hilbert category, namely one assigns to each symplectic manifold X the Hilbert space L^{2}(X), where one takes L^{2} with respect to the Liouville measure. Every symplectic map yields a unitary operator in an obvious way. This is called prequantization. The problem with it physically is that a one-parameter group of symplectic transformations generated by a positive Hamiltonian is not mapped to a one-parameter group of unitaries with a positive generator. So my conjecture is that there is no "positivity-preserving" functor from the symplectic category to the Hilbert category.)
Second quantization is the attempt to get from a quantum description of a single-particle system to a quantum description of a many-particle system. (There are other ways to think of it, but let's do it this way.) Starting from a Hilbert space H for the single particle system, one forms the symmetric (or antisymmetric) tensor algebra over H and completes it to form a Hilbert space K, called the bosonic (or fermionic) Fock space over H. Any unitary operator on H gives a unitary operator on K in an obvious way. More generally, one has a functor called "second quantization" from the Hilbert category to itself, which sends each Hilbert space to its Fock space, and each unitary map to an obvious unitary map. This functor is positivity-preserving. (All the weird problems with negative-energy states of the electron, Dirac's "holes in the electron sea," and such, are due to thinking about things the wrong way.)
By the way, you might enjoy thinking about what happens if you iterate second quantization. Give up? My, that was quick! Well, try reading my stuff on "nth quantization".
Now, an interesting object in physics is Minkowski space. We can imagine a category Mink which has only one object - Minkowski space! And whose morphisms are the Poincare transformations (i.e., rotations, translations, Lorentz transformations, and composites thereof)! Then one can imagine a natural transformation from Minkowski space to the category Spin with one object, the space of spinors (fancy for 4-tuples of complex numbers), and morphisms given by the representation of the Poincare group on this space. Then what expresses the principle of relativity most precisely is that the value of any observable, e.g. a spinor, must define a functor from Mink to the relevant category, in this case Spin. We can also express the principal of general covariance and the principal of gauge-invariance most precisely by saying that observables are functorial. So physicists should regard functoriality as mathematical for "able to be defined without reference to a particular choice of coordinate system."
Physicists are entitled to regard this as a bunch of high-falutin' abstract nonsense until I give them an example in which "functoriality" really has more to it than just covariance under a group action. The point is, that a category is really a generalization of a group. Group representations are really only a special case of category representations. This idea was sold to me by Minhyong Kim. He said: "Eventually people will see that group representation theory is not such a big deal; what really matters is representations of categories." At first I thought he was just trying to sound slick (he always goes for the most abstract and elegant viewpoint). But then I wound up needing category representations in my own work on quantum gravity.
In article <BsMHq1.JpH@cs.psu.edu> sibley@math.psu.edu writes:
>Speaking of groups and categories, I have always liked the category >version of the definition of a group: > > A group is a category with one object in which all the morphisms > are isomorphisms.
A representation of a group, if we think of a group as a category as Sibley suggests, is just a functor from that category to the category Vect of vector spaces. So we can define a representation of a category to be a functor from that category to the category of vector spaces.
An example of an interesting category with interesting representations is the category of tangles! Tangles are like braids but the strands can double back on themselves and there can also be closed loops. The objects in the category Tang are {0,1,2,...} and the morphisms in Hom(m,n) are (isotopy classes of) tangles with m strands going in and n strands coming out. A picture is worth a thousand words here. Here is an element of Hom(2,4):
| | \ / \ / \ /\ / \ / \ | \ / \ | \ | | / \ |
2 in, 4 out! Here is an element of Hom(4,0):
| | | | \ / \ /\ / \/ \ \ / \/ \ \____/
4 in, none out! We can compose these morphisms to get a morphism in Hom(2,0):
| | \ / \ / \ /\ / \ / \ | \ / \ | \ | | / \ | | | | | \ / \ /\ / \/ \ \ / \/ \ \____/
To be precise, a tangle is a 1-manifold X with boundary embedded in [0,1] x R^{2}, such that boundary of X is mapped to the boundary of [0,1] x R^{2} and such that X intersects the boundary of [0,1] x R^{2} transversally. We also assume that the points in the boundary of X get mapped to certain "standard" points (0,x_{i}) and (1,x_{i}) in the boundary of [0,1] x R^{2}, so we can compose tangles by gluing them together as in the picture above. There is thus a category whose objects are {0,1,2,....} and whose morphisms Hom(m,n) are isotopy classes of tangles with m boundary points in {0}xR^{2} and n boundary points in {1}xR^{2}.
It turns out that one can get a representation of the category of tangles from any finite-dimensional representation of a semisimple Lie group. This construction is due to Reshetikhin and Turaev and involves quantum groups. The subject got started here:
For more on the category of tangles, click here.
Okay, now back to MacLane's cryptic remark. What's a natural transformation?
Well, natural transformations are things that go between functors. Suppose we have two functors F and G from the category C to the category D. Then a natural transformation N from F to G assigns to each object X in C a morphism N(X): F(X) → G(X) such that this diagram commutes:
F(X) -F(f)-> F(Y) | | N(X)| |N(Y) v v G(X) -G(f)-> G(Y)In other words, this equation holds:
G(f) N(X) = N(Y) F(f)
An example would be "abelianization", which maps a group H to the abelian group H/[H,H]. If F were the fundamental group and G were the first homology group, we could say that abelianization is a natural transformation from F to G. This example is a bunch of mumbo-jumbo unless you know some algebraic topology, but I should be able to think of a more physical one.
If you want to get into deeper waters, think about this question:
What sort of thing is the "category of all categories"?
It turns out to be, not just a category, but a 2-category. That means that in addition to objects and morphisms, it has "2-morphisms", that is, morphisms between morphisms. To see how this goes, let's call the 2-category of all categories "Cat". Then the objects of Cat are categories, the morphisms of Cat are functors, and the 2-morphisms are natural transformations!
Let me just say a bit about where things go from here. First of all, it turns out that we can keep playing this game ad infinitum. We can define a notion of "n-category" having objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms... and it turns out that "category of all n-categories" is really an (n+1)-category.
Just as we can talk about a representation of a category, that is, a functor to Vect, we can talk about a representation of an n-category. This is the same as an "n-functor to nVect", where nVect is the n-category of "n-vector spaces". While this doubtless may seem like insane generalization for its own sake, this idea shows up naturally in topological quantum field theory and string theory. So, it could turn out to be important in physics. At the very least, it's beautiful stuff.
For more about n-categories and physics, try the following papers, listed roughly in order of increasing difficulty - but with a heavy bias towards ones I helped write!
If the above papers are too technical for you, you might try reading week49 of This Week's Finds for a quick overview of n-category theory, or the series starting with week73 for a more detailed, but still light-hearted, exposition. To learn what's going on now with n-categories and physics, try Urs Schreiber's blog.
You might also try my quick introduction to a rather different aspect of categories, namely topos theory.