What are the applications of category theory in physics?

[Haelfix]

Im a mathematical physicist, and lately I keep reading papers that throw this jargon around, and its beginning to bother me that I don't know anything about it, it feels like a gap in my knowledge.

I was trained as a mathematician in undergrad so of course it is somewhat familiar to me, proffessors used to throw the lingo around to act smart.

However, it never seems like it buys you much, at least in physics. It just looks like a more complicated (albeit perhaps more elegant) exercise in abstraction.

So 2 questions. Does anyone know of useful nontrivial results that were only apparent with category theory in physics, and does anyone have a good reference/weblink that gives an easy introduction for an idiot =)

 

[matt grime]

Hmm, this is one of my bug bears. I don't know why category theory always gets stick but other formalisms such as group theory or differential manifolds are learned by physics students without a murmur.


the point about category theory is that is the abstraction that makes it powerful and shows us how to use results in algebra in topology or geometry.


the current vogues in categorical physics are two fold - one is n-categories (Baez, this week's finds passim) where homotopy like results are used to describe paths and paths of paths and paths of paths and so on of whatever you care about. the second are triangulated categories which are the natural language of vector bundles et cetera. strings and branes are the unifying theme.


it's still a new language, and it remains to be seen just what can be done with it.

if you want a purely categorical theorem about non-categorical objects then Broue's theorem is an example - the derived categories of principal blocks of groups with abelian sylow subgroups are derived equivalent (there is a categorical equivalence) with the principal block of the normalizer of the sylow subgroup. this categorical result explains many things that are otherwise just coincidences.

 

[robphy]

A while back (and when I had the time), I wanted to learn category theory to understand the structures underlying physical theories.

Here's a list of references that looked helpful at the time (although I haven't had the time to get too much past the first few pages on the subject). I would be interested on opinions on these... someday, I'd like to return to this subject.

Geroch, Mathematical Physics,

http://www.worldcatlibraries.org/wcpa/top3mset/d1118c31ac33b711a19afeb4da09e526.html

Lawvere and Schanuel, Conceptual Mathematics : A First Introduction to Categories,

http://www.worldcatlibraries.org/wcpa/top3mset/89555bba70a7ebe1a19afeb4da09e526.html

Mac Lane, Categories for the Working Mathematician,

http://www.worldcatlibraries.org/wcpa/top3mset/c881342209848a32.html

Arbib and Manes, Arrows, structures, and functors : the categorical imperative, http://www.worldcatlibraries.org/wcpa/top3mset/596f4e7c805bc7c3.html

I've noticed that Computer Scientists are starting to study Category Theory... but I'm not sure why.

Recently, I found a collection of online references on
http://us.geocities.com/alex_stef/mylist.html (see "Category Theory")
http://plato.stanford.edu/entries/category-theory/
http://www.math.mcgill.ca/triples/
http://www.mta.ca/~cat-dist/categories.html

And, of course, as mentioned above, there is John Baez's http://www.math.ucr.edu/home/baez/categories.html

 

[Hurkyl]

I think Computer Science interests are based on the applications of category theory to logic rather than topology.

 

[mathwonk]

The elementary part of category theory is merely a point of view, and it is related to physics at least to me.

it is that one should do things in ways that are as natural as possible and as coordinate - free as possible.

and also that one should be aware of, and sensitive to, how much structure you are using at the time.

for example to do problems that deal only with smoothness and not with length, it is more natural not to use a Riemannian metric on your manifold, since it imposes more structure than the situation calls for.

it is also preferable to understand that a tensor is a sort of multilinear operator, and not simply a mass of indices.

The first definition of any importance in category theory is that of a morphism ("map that respects the given structure").

the second definition is that of an isomorhism, ("map that leaves the given structure unchanged")

i.e. suppose we are in the world of smooth structures (calculus on differentiable manifolds).

then a morphism is a differentiable map, and an isomorphism is a differentiable map witha differentiable inverse.


then the next definition is that of an operation (called a "functor")that does two things:
1) it transforms an object with one type of structure into an object with a different type of structure .

for example the operation of taking tangent vectors at a point, transforms a manifold M and a point p on M, into a linear vector space Tp(M) the tangent space to M at p.

2) it also transforms a morphism between two objects with the first type of structure into a morphism between the two corresponding objects with the second type of structure.

example, a smooth map between two manifolds f:M-->N, taking p to q, is tranformed by the tangent space functor (or the derivative functor), into a linear map df(p) from Tp(M)-->Tq(N).

thus the derivative functor really has two parts:
1) change a smooth space into a linear space

2) change a smooth map into a linear map.


now functors are required to have two simple properties:

i) they respect compositions, (e.g.: chain rule dp(gof) = dg(q) o df(p)).

ii) they respect identities: if f = id:M-->M, then df(p) = id:Tp(M)-->Tp(M).

It follows iimmediately from the definition of isomorphism that all functoirs take isomorphisms to isomorphisms.

thus the derivative of a smooth isomorphism between two manifodls must be a linear isomorphism between their tangent spaces.

In particular, since linear isomorphisms preserve linear dimension, smooth isomorphisms also preserve the dimension of the tangent space, hence also the dimension of the manifolds.

i.e. the fact that the derivative of an invertible differentiable map is non zero is almost a tautology.

the proof that the dimension of continuous manifolds is preserved by continuous isomorphisms is harder, but similarly uses an appropriate functor, e.g. the local homology group. i.e. the puncturerd nbhd of a point on an n manifold is homotopic to an (n-1) sphere hence has reduced homology in only dimension n-1.

we think this is harder because we learn calculus before we learn homology. but remember how hard it was to learn calculus? maybe calculus is no more elementary than homology. [this suggests already to me that local homology may play a role in generalizing calculus techniques to the continuous case. see what suggestive power category theory has?]

so one advantage of category theory is to render trivial, a large volume of material that used to be thought to have some content. another is to suggest generalizations and conjectures.


Another game in category theory is rephrasing lots of common definitions and concepts using only "commutative" diagrams of maps. this results in seeing many analogies between previously unrelated ideas.


for instance a "product" of two objects X,Y is another object Z equipped with a pair of morphisms Z-->X, Z-->Y (think projections), such that a function W-->Z is a morphism if and only if both composed functions W-->Z-->X and W-->Z-->Y are morphisms.


an object Z is a "sum" of X and Y if there are given morphisms X-->Z and Y-->Z such that a function Z-->W is a morphism if and only if both composed functions
X-->Z-->W and Y-->Z-->W are morphisms.


then one notes a subtle shift in emphasis from the traditional definitions say in set theory where a product is "defined" as a set XxY of ordered pairs (s,t) with s in X and t in Y.

In the new world, that is no longer a "definition" but a "construction", presumably one of many possible (but isomorphic) ones.

[what is a construction of the "sum" of two sets?]

I.e. now instead of presenting a construction of a desired object, instead one writes down how it should behave under maps, and then poses the problem of constructing it, i.e. proving it exists.


so, category theory imposes a new way of thinking, or of looking at problems.

[I realize that in writing this, I have found one of the reasons I do not comunicate well with some physicists here: I think in categorical terms and some physicists do not.]

e.g. if we abandon the idea that an arrow is a function and write a-->b for a, b integers to mean a divides b, then we see that if z is the l.c.m. of a,b, then we have

a-->z and b-->z and also that z-->x if and only if a-->x and b-->x. i.e. z is the categorical "sum" of a and b!

what do you think the categorical "product" of a,b, would be in this sense? prove it.

i am tired now.

 

[matt grime]

another favourite categorification is that a group is a category with one object and all morphisms isomorphisms.

category theory also gives us a way to think about limits - we can describe ('infinite' in some sense usually) things as certain kinds of limits of 'finite' things, eg a vector space can be thought of as the limit of all its finite subspaces. what properties of the finite objects does the infinite one possess?

if you like think of category theory as a babelfish translator - who'd've thought that lcm's and unions or gcds and products are the same (well, actually if you think about ring theory the second one is clear - i guess we're forming a functor from the diagram for product on the catergory of ideals in Z were morphisms are inclusions - nice example).

 

[mathwonk]

Intro to categories, part II, limits:

matt's mention of limits refers to a generalization of products and sums.

I.e. notice that the exercise above was to show that the "sum" of two sets is their disjoint union. but what about their actual union? is that a sum of some kind?

It is a sum "with amalgamation", i.e. with overlaps between the two summands.

so here the data consists of the two sets S, T plus their intersection I, as a further set.

then there are inclusion maps from I into each set, I-->S, and I-->T. then a sum of S and T that respects those inclusions would be an object X plus, let's see, a pair of maps I guess S-->X, and T-->X. as before, but now also we want the two compositions I--S-->X and I-->T-->X to be equal.

this is precisely the case if X = S union T, and the maps S-->X and T-->X are both inclusions.


In this case X is a sort of limit of the system of maps {I-->S, I-->T}. Notice there is a family of sets here, I,S,T, and two sets are connected by a map if and only if there is an inclusion between them. Thinking of set inclusion as a partial order on all sets, this means the maps are indexed by a partially ordered set.

To push the same idea further we just change the partially ordered index set to an arbitrary one.

I.e. suppose we have a whole family of maps {Ai-->Aj} indexed by some partially ordered set of indices {i}.

Then a limit would be an object X with a map Ai-->X from each of our sets, plus the requirement that whenever two sets Ai,Aj are connected by a map Ai-->Aj, i.e. whenever i <= j (or vice versa, I can never remember),

then the composition Ai-->Aj-->X, equals the map Ai-->X.

gee, somehow i made this l;ook a little different from the model case above of unions, but so what.

you wil notice that if the Ai are all sets and the maps Ai-->Aj are all inclusions of subsets, that then the union X of all the Ai satisfies this rule, where all the maps Ai-->X are inclusions.

this is called "direct limit", and the analogous generalization of a product is called an "inverse limit".

check that when the Ai-->Aj are again all inclusions that the inverse limit is the intersection of all the Ai. (with some appropriate conventions on the partial order).

Now to really go round the bend, if I is any category, i.e. any collection of all objects having the same structure, and their morphisms, and if C is any other category, there is for each... [good heavens i can't even remember this myself, i'll have to read my own book...ok i think i got it.].

OK, note that a category is like a partially ordered set. i.e. some objects have maps between them, some do not, and when there are maps X-->Y, and Y-->Z, their composition gives a map X-->Z. (transitivity of ordering)

A system of objects indexed by a partially ordered set is just a functor on this category. so we want to be able to define the limit of any functor defined on a partially ordered set, so we might as well define the notion of limit of any functor defined on any category. {After all part of the fun of category theory is to make it so general the only application is to confuse and frighten the uninitiated.]

So let I and C be categories and consider the category of all functors from I to C, called Fun(I,C). [Don't tell anyone, but this is to be thought of as the collection of all systems of elements of C, indexed by elements of the partially ordered set I.]

Now notice there is a functor from C to Fun(I,C) called the "constant functor", which takes an element X in C to the functor sending every element of I to X. I.e. this is the constsnt indexed system where all the indexed objects are the same object and all the maps are the identity isomorphism of that object.

This defines a kind of "stupid functor" c:C-->Fun(I,C).

(using the word "stupid" is also calculated to frighten away more of the people still hanging around hoping to understand something.)

Now a direct limit construction is an "adjoint" of this constant functor,

i.e. it is a functor

lim:Fun(I,C)-->C, such that whenever {Xi} is a functor in Fun(I,C), then

lim({Xi}) is an object X in C, such that for every element Y of C,

the following two families of morphisms are equivalent:

Hom({Xi},c(Y)) = Hom(X,Y).

i.e. there is a family of morphisms Xi-->X, respecting the compositions Xi00>Xj,

hence anyone morphism X-->Y, yields the compositions Xi-->X-->Y, hence a compatible family of morphisms Xi-->Y, which is a 1-1 correspondence between the morphisms in the sets above.

I.e. "compatible" families of morphisms from the system {Xi} to Y, is the same as one morphism from the object X-->Y. That is what it means to say X = lim({Xi}).

whew! I have never used this definition except to try to intimidate people, but maybe someone else has another use for it. (Someone who understands it better than I do for example, could use this opportunity to expose me as a categorical fraud.)


[Obviously I like this subject, but also regard it somewhat lightly, as a bit of a joke, but one with significant consequences.

I.e. the subject is actually easy, but has been made to look hard in the literature, and I wish to reveal that this should be poo - pooed as much as posible in order to maintain enough dignity to actually master it.

as sylvanus p thompson so famously said of infinitesimal calculus:
"what one fool can do, another can".

 

[mathwonk]

monsieur haelfix, is this of interest? otherwise it will stop here.

 

[Haelfix]

No it looks very interesting, if still a bit new and raw to me (it feels like homology theory as you pointed out.. inclusion maps and what not). I am trying hard to enjoy a 3 day vacation without indulging my obsession with physics/math, but I am definitely going to devote a few hours on tuesday and probably more beyond. I'll post further questions/problems when I feel a little more informed so as not to completely bore everyone with trivialities.

I really need to first see how to map set notation to category notation, and how far concepts in the first (which I know well) are subsumed by the latter, in so far as they are still *good* notions. You pointed out a good deal with the inclusion maps (it makes good sense) but some of the tricky bits of set theory (like the axiom of choice) seem to be only specific cases here.

 

[matt grime]

in categorical terms we can entirely forget set theory, though i don't think that a good idea.

the difference you may have to come to terms with is that although the objects in a category are sets usually, we shouldn't think of these sets as having elements.

in category theory we do not talk in general about injections or surjection. instead we try to find a definiton of injection that doesn't involve comparing f(x) with f(y). that notion is monomorphism. f is mono if fg=fh implies g=h so called left cancellable property. surjections are replace with epis and are right cancellable.

in the category of SET these are just inj/surj but in other categories they are not necessarily. for instance in RING the inclusion of Z to Q is an epi that is not surjective (any ring hom from Q to anything is uniquely deteremined by where it sends the integers).

in lots of physically interesting cases it doesn'te even make sense to thik of morphisms as functions like f(x); in particular string theorists like to know about derived categories and there it is difficult to visualize the maps if you insist on thinking of f(x).

if you like category theory is about the structure of the objects rather than the object's elements.

for various purposes, though, we tend to require that certain things are sets, mainly that the morphisms from X to Y are a set, or that there is a set of isomorphism classes. this restriction is mainly to force the things we construct, like limits, to lie in the same universe as the original objects.

here's an example to shyow why it's important to have restrictions on the objects - it isn't a set theoretic one, but i can't think of a simple one where we *need* to have sets of things lying around; trust me they are important esp. in the physics format. take the colimit of mathwonk. there is an obvious diagram in the category of finite dim vector spaces, the inclusion

R to R^2 to R^3 ...

the colimit of this does not exist in fin dim vect spaces. the colimit passing to all vectro spaces is just a countable direct sum of R.

in general for module categories (or abelian categories as it happens) colimits only exist if the category has infinite direct sums since given

X_1 to X_2 to X_3...

the colimit is the cokernel of a map from

$$\oplus X_ i \to \oplus X_i$$

the map is easy to describe. if d stands for the generic map from X_i to X_{i+1} then the map is "Id - d" so it sends x in X_i to x-dx in X_i + X_{i+1}. this is an injection, so we can form the short exact sequence and get the colimit as the 3rd object.


the set theoretical worries are generally when we try to make a universal object, and the construction involves taking a direct sum over "all objects" or over the isomorphism classes of objects at some stage - this won't in general exist "in our universe" if this index isn't a set.