So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of mathematics. Of course, the precursor to this was the work done by Devinatz, Hopkins, and Smith in classifying the thick subcategories of the stable hootopy category of (finite) spectra. They famously used this classification to prove the periodicity theorem, and I can see why it is helpful: the classification theorem reduces the periodicity theorem down to the (still nontrivial!) task of finding a single type $n$ complex with a periodic self-map. I'm sure similar uses have been found for the other classification theorems, but I am left to wonder, more generally:
What kinds of problems are made simpler with a classification theorem? What questions does it answer?
I'm looking for some general heuristics here. They should satisfy the following conditions:
They should apply to theorems already proven by classification theorems; examples and references would be lovely here!
These heuristics should come with some sense of why one would think to use the classification theorem in this way. For example, I can see how the classification theorem makes the periodicity theorem manageable to prove, but why would one think to use it in the first place?
This last one is more of a throw away or a bonus, but it's worth a shot: If there are any areas of mathematics, or open problems that you think are begging for a classification theorem type application then please share! It would be a useful test of the proposed heuristics if they are able to predict the solution of a problem that has not been solved...
Basically I'm looking for some intuition here. My logic being: if we know more about what kinds of questions a classification can answer then we will know more about the information contained in a classification. This, in turn, may provide clues for how to compute or construct such a classification (which is, of course, the next step in Balmer's program).
P.S. I've tagged the areas that I know of with classification theorems. If I'm forgetting some, do remind me in the comments Looks like there's a limit on tags :).)