# What classification theorems have been improved by re-categorizing?

Many classification theorems (e.g. of the finite subgroups of $SO(3)$, or the finite-dimensional complex simple Lie algebras, or the finite simple groups) have some infinite lists, plus some "sporadic" or "exceptional" examples.

It was opined in Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3... that any such theorem only reflects our current knowledge of the subject. One might hope that better understanding would lead to an alternate way of slicing up the set of examples, with fewer sporadic cases.

Are there actually examples of this happening? I want a case where the initial classification is actually complete and correct; it's only our human description that has improved.

Two non-examples: (1) As I understand it, Suzuki found an infinite family of finite simple groups, that we now regard as twisted Chevalley groups for $G_2$ and its outer automorphism in characteristic 3. But that was before the classification was complete. (2) Killing had two root systems that turn out to both be $F_4$. So he wasn't quite correct.

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"Examples of theorems with proofs that have improved" is in the same spirit, but I'm asking for an improved statement. mathoverflow.net/questions/95837/… –  Allen Knutson Nov 20 '12 at 14:49
I think there has to be an example involving Galois theory and descent: blodiglums over $K$ are classified by blodiglums over $\overline{K}$, together with an action of the absolute Galois group $\mathrm{Gal} \overline{K}/K$. However, perhaps due to my almost complete ignorance of history of mathematics, I haven't been able to come up with an example of "blodiglums". –  Arend Bayer Nov 21 '12 at 16:51
Would Grothendieck's reformulation of the classical Galois correspondence between extension fields and subgroups to actions of the Galois group and etale algebras count? –  Benjamin Steinberg Nov 21 '12 at 17:06
Community Wiki? –  Todd Trimble Nov 21 '12 at 19:29
Did Grothendieck's reformulation really simplify the original? –  Allen Knutson Nov 22 '12 at 1:01

"Classification" might be a very strong word for this example, but I think real quadratic polynomials underwent something like the development you describe. They are classified by their discriminant: positive for two distinct roots, zero for a double root and negative for no roots ("three families"). The generalization of complex polynomials and roots simplifies this: discriminant zero for a double root and non-zero for two distinct roots ("two families"). When you specialize this to real polynomials you end up with what you started with, with the added bonus of two complex conjugate roots instead of none in the case of a negative discriminant.

Surely this is only a toy example, but it does show that broadening the viewpoint can reduce the number of cases to consider.

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Nice. This makes me think of the classification of conic sections into parabolae, hyperbolae, ellipses, and pairs of lines, that becomes only two cases (smooth conic vs. pair of lines) once one projectively-completes. –  Allen Knutson Nov 22 '12 at 1:00

I (think) that the classification theorem of Chudnovsky, Robertson, Seymour and Thomas for Berge graphs qualifies. Namely, their strategy for proving the Strong perfect graph theorem was to prove the following theorem.

Theorem. Let $G$ be a Berge graph. Then at least one of the following holds:

1. $G$ (or its complement) is bipartite.
2. $G$ (or its complement) is the line graph of a bipartite graph.
3. $G$ is a double-split graph.
4. $G$ (or its complement) admits a proper 2-join.
5. $G$ admits a balanced skew partition.
6. $G$ admits a homogenous pair.

For definitions of these terms, see the proof of the Strong perfect graph theorem.

It turns out that this structure theorem is sufficient to prove the Strong perfect graph theorem since one can prove that bipartite graphs, line graphs of bipartite graphs and double-split graphs are perfect and a minimal counterexample to the theorem cannot satisfy (4), (5) or (6).

To actually answer the question, Chudnovsky later proved that (6) can be dropped from the above structure theorem. The proof of the improved statement is over 200 pages, so this is a certainly a non-trivial re-categorization. I do admit it is a bit unsatisfying that this re-categorization is just a proper subset of the original categorization.

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My understanding of algebraic geometry is woefully inadequate, but here goes.

My example is Borisov-Chen-Smith's definition of toric Deligne-Mumford stacks in terms of stacky fans. In other words, essentially a definition by construction. Fantechi-Mann-Nironi then gave a definition of toric DM stacks as closures of orbits of "stacky tori." I believe they also showed that different stacky fans may give rise to isomorphic toric DM stacks.

I don't think the Fantechi-Mann-Nironi definition is the ultimate answer to the question of what a toric DM stack is. This is because on the symplectic side there are toric DM stacks whose generic stabilizer groups are non-abelian (see arXiv:0908.0903v2 [math.SG])

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