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Over on the nCafe some Thomas asks
Conc. “Triangulated categories in algebraic geometry”, was Grothendieck not unsatisfied with triangulated categories and suggested “derivatuers”? How is the current state of those and which applications could they have in algebraic geometry?
I can’t reply there for some reason. My speculation is that the number of hyperlinks in my reply is triggering the spam filtern. So I post my reply here and point Thomas to it.
Here goes:
Thomas said:
Conc. “Triangulated categories in algebraic geometry”, was Grothendieck not unsatisfied with triangulated categories …
Yes, so an triangulated category is the decategorification of a (oo,1)-category that is stable – stable under forming loop space object and suspension objects, that is. The shift functor in the triangulated category is the decategorified shadow of these canonical suspension operations. The “triangles” in the triangulated category are the decategorification shadow of the fiber sequences in the (oo,1)-category.
As always, applying decategorification is a bad idea and destroys lots of data. That’s why everybody hates it. (But just like other destructive deeds, everbody nevertheless often does it.)
On the other hand, for a long time various notions of enhanced triangulated categories have been used, which are now understood to be equivalent to the full (oo,1)-categorical incarnation.
…and suggested “derivatuers”?
That is not specific to triangulated categories, but is generally a method for handling (oo,1)-categories. See derivator.
How is the current state of those and which applications could they have in algebraic geometry?
The current state is that algebraic geometry is completely undoing all secret decategorification and therefore now calls itself derived algebraic geometry. Stable (oo,1)-categories (or one of their enhanced triangulated incarnations) are everywhere.
Although derivators do work pretty well for dealing with the stable/triangulated case, and I expect that it was that case that Grothendieck etc. had primarily in mind when developing them.
Urs said:
an triangulated category is the decategorification of a (oo,1)-category that is stable
Not every triangulated category is topological! That is not every Verdier triangulated category comes from a category enriched in spectra. Hence this is not true in general. There is a hierarchy
Puppe triangulated category (most general) Verdier triangulated category (require octahedron) strongly triangulated category (higher analogues of octahedra, Neeman, Maltsiniotis) topological triang. cat. // coming from spectra enriched/stable category algebraic triangulated categories //coming from dg or AS infinity
One coudl add between algebraic and topological the hypothetical case of algebraic over F_1.
An intermediate notion, which contains a good deal of this information, is a derivator.
what is strange as we agreed that most likely the derivator and stable infinity category in the case which is related to enhancing the usual triangulated categories give the same information.
It woudl be good to have separate entry topological triangulated category but I am not competent to write it.
I thought Urs meant that most triangulated categories arising in practice are the homotopy categories of stable -categories, which I think is true.
most likely the derivator and stable infinity category in the case which is related to enhancing the usual triangulated categories give the same information.
At least in the locally presentable case, the paper of Renaudin that I mentioned in my latest cafe post suggests that the derivator contains all the information of the -category, in the sense that the latter can be recovered from the former up to equivalence. However, even in that case I think it’s still fair to say that the -category contains “more information.” For instance, I see nothing suggesting that derivators can “see” functors between -categories which don’t have adjoints.
Yes, I meant those arising in practice.
Does anyone have an example of a triangulated category that is being used for anything which is not of this kind?
I would definitely like to hear from an expert on triangulated categories, whether or not people actually use any triangulated categories that don’t arise from stable -categories, or whether they are just curiosities.
If nobody here knows, this might be a good question for a blog post.
Because either one takes the view that
[…] they are just curiosities
or one takes the view that some definitions of triangulated categories were simply wrong in the sense of: not a good definition. I am inclined to regard this situation as an example of where (higher) category theory helps with finding the right definitions.
I don’t think the view that “they are just curiosities” and the view that “the definition was wrong” are really much different if at all; by “curiosity” I meant to indicate that their existence is an accident due to the definition not being right.
Sure, right.
Well there are situations where there is some natural construction and one can prove that one can construct from it a triangulated category but can not prove more. So many theorems will fail if we restrict a notion of triangulated category, and sometimes one even can prove only Puppe triangulated category. On the other hand, there are some variants of the notion of derivator as well. For example, Kaledin has a major unpublished theorem on algebraic K-theory which amends the problem Neeman had with a counterexmaple for the conjecture that the K-theory of a scheme (under some weak assumptions) is captured by the derived category of coherent sheaves; Kaledin takes an interesting modification of a derivator (the modification is in the style of Grothendieck school: one requires some required functors just for certain class of maps/diagrams) and then he has a theorem in spirit of the original conjecture of Neeman. He gave a talk on this discovery in Moscow, maybe a year ago, but there is no writeup. So it is difficult in present state to give so strong statement as Urs is giving on what is stronger and what is not.
Well there are situations where there is some natural construction and one can prove that one can construct from it a triangulated category but can not prove more.
Can you give some examples?
For example I have seen a construction of Pirashvili (it must be somewhere online the article) leading to Puppe triangulated categories but it is open problem if it is possible to enhance the construction even to get he Verdier triangulated category.
I was hoping for a little more information than that. A brief Google search doesn’t turn it up.
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