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I started typing into length of an object when I felt that we had an entry on this already somewhere. Where?
I didn’t think we had such an entry, and was thinking about writing up something not very long ago, but never got around to it. I think I had in mind something lattice-theoretic (e.g., geometric lattices).
Searching for “jordan-h”, we have 8 pages now using Jordan-Holder (with and without umlaut) as a qualifier of sequence, theorem, or coefficients.
We also have object of finite length.
And composition series should be integrated too.
Thanks!
I made “object of finite length” a redirect to length of an object. I left composition series as is, since I gather one tends to say this more for (non-)abelian groups, while “length” is more meant to be in abelian categories. (?)
What are those 8 entries though that mention “Jordan-Hölder” or “Jordan-Holder”? I asked Google, and it didn’t give me any.
nLab’s own search is good for strings. Aside from appearances in titles of references on some pages, there are 5 we haven’t mentioned:
I have added to length of an object the Relation to Schur functors for the case of tensor categories in characteristic zero.
So David Jordan here says that such tensor categories which violate the condition that for each object there is a number such that are “obscenely large”.
Can we make this more precise? What is pathological about tensor categories that violate this condition? I mean, it is clear that this condition is one that fits our intuition of dimension of an object, but how can we make more precise the idea that it makes sense to disregard in practice categories that violate this condition (which I suppose is what is meant by them being “obscenely large”)?
In physics terms is this something like:
Given a composite system, then there’s a limit to the number of resulting particles from smashing copies of it together. Growth is at most exponential.
re #6, oh, I see, so the entry was being requested but didn’t actually exist yet. So I started a stub Jordan-Hölder theorem
re #8, sure, the intuition is clear. But can we make more precise mathematically what is “obscence” in the situation where this intuition fails.
So first of all: what are examples of tensor categories that fail this condition?
Does anyone know the right generality for Jordan-Hölder theorem? Omega-group claims it holds for all of its versions. Are we in protomodular category territory?
I guess the answers are in this MO answer.
So I’ve added in the connection to homological category.
Thanks for the pointer to Omega-group.
I don’t know why there the link to “Jordan-Holder threorem” does not work. The hyphen may have triggered the hyphen bug, but I copy-and-pasted the requested link text as a redirect into the JH-theorem entry, which usually makes that bug go away, but in this case to no avail.
You’d missed the ’d’ in Jordan in the redirect. Fixed now.
Oh, how silly of me. Thanks for catching this!
Re #7, p. 302 of Tensor Categories looks useful. The condition is related to having a fiber functor to .
That’s how Deligne proves the Deligne theorem on tensor categories. He first shows that the condition of subexponential growth of length implies that is annihilated by some Schur functor, then he shows that the latter implies the existence of a fiber functor to super vector spaces, then he finds that the automorphisms of that form an affine algebraic supergroup, then he concludes with his general Tannaka reconstruction theorem.
But what I am trying to get a better grasp of is how to see that the assumption of subexponential growth, which is the basis for Deligne’s theorem, is a natural one. I certainly find it natural intuitively, but I am hoping one could expand a bit more on how the failure of sub exponential growth of length is “obscene” in a mathematical sense.
But that textbook account is neat. I have been adding links to it. Also, the full pdf is freely available: pdf.
Perhaps “obscene” wasn’t a great choice. In that book, they’re saying it’s the slowest you can hope for. already has exponential growth.
So we see a stark contrast with other fields of mathematics…where exponential growth is viewed as fast. In the theory of tensor categories, this means the slowest possible growth, and we will see that things can get worse.
Oh, now I see that counter-examples are discussed in section 9.12 of that book.
There’s this from Matiyasevich in a different context:
My first impression of the notion of a relation of exponential growth was “what an unnatural notion”, but I soon realized its important role for Hilbert’s tenth problem.
There is also, of course, work of Gromov on groups of polynomial growth.
These are perhaps not directly relevant, but show the yoga of such growth rates is indeed present.
One person’s obscenity is another’s point of interest:
…the fast-growing categories…are not pathological examples, but rather are glimpses of a new and largely unexplored world of combinatorics of sets of “non-integer cardinality” and linear algebra of vector spaces of “non-integer dimension”. (remark 9.12.15)
In Proofs and Refutations, Lakatos wrote about this ’monster-barring’ of objects, using examples from 19th century topology taken to be deformed in some way.
re #20, strange, I would have thought that the condition is certainly natural. It simply says that the appropriate concept of dimension of objects behaves as we expect dimesions to behave.
re #21, ah, thanks for this pointer, that’s useful.
re #18 (cross-posted, I was looking up the Hilbert’s Tenth material)
ah, I see, that makes sense.
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