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The following old material was sitting at triangulated category and was labeled “discussion”. I am hereby moving it from there to here. Probably some of it deserves to be merged into the entry in some form, but under a headline “history”.
[begin forwarded discussion]
The original definition of triangulated categories is apparently due to Verdier, who developed the theory upon guidelines by Grothendieck; Dold and Puppe developed independently a version without octahedron axiom with motivation in algebraic topology. In the manuscript Pursuing Stacks, Grothendieck mentions that the usual definition of triangulated categories and the corresponding derived categories seemed to be inadequate for some of the developments that he wished for. He also says something to the effect that he had tried to interest various of his ex-students in doing a thorough treatment of the ideas, which he considered to be necessary for future development, and which he then proceeds to sketch out.
+–{+ .query} Zoran Skoda: I am not quite sure if this is entirely correct. Grothendieck indeed wanted more flexibility in homotopical algebra and went to develop these things; but if one talks only very specifically about the concept of triangulated category itself (not wider context) than the main complaint of everybody was about the crudeness of localization at quasiisomorphisms; the thing which for example Drinfel’d’s “quotients of dg-categories” paper successfully rectifies (and then again Lyubashenko in quotients of $A_\infty$-categories). =–
That led to the theory of derivators, where the idea is that in addition to looking at a basic category of ’things’ such as chain complexes, you should also look at all categories of diagrams of such things, and the derived / homotopy Kan extensions between the corresponding derived categories that correspond to a change of the indexing category. The basic idea behind this was also explored slightly later by Alex Heller (1988). See the references on the pages derivator, pointed derivator, and stable derivator.
[end forwarded discussion]
Urs, we had earlier convention to leave a link to the archived discussion at the previous place in the entry. This way it would not be lost in the forum.
I put in a link.
Zoran,
isn’t it easy for you to modify the text such as to address your query-box remark, and then insert is properly back into the entry? It does not seem to be a controversial point to me.
I do not open all threads on nForum. If you are doing removal without a link, some will slip even when I can manage, because I do not notice the thread. I certainly did few times such corrections quietly in fact. But the person who is already reorganizing the page and knows where the things stood up will also do better organizing where to add links. So it is better that I leave a remark to you.
These days I can hardly open 20% of threads in nForum, and will diminish in days to come.
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