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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.
I edited the idea section a bit. It now reads:
Any (infinity,1)-category $C$ can be flattened, by ignoring higher morphisms, into a 1-category $ho(C)$ called its homotopy category. The notion of a triangulated structure is designed to capture the additional structure canonically existing on $ho(C)$ when $C$ has the property of being stable. This structure can be described roughly as the data of an invertible suspension functor, together with a collection of sequences called distinguished triangles, which behave like shadows of homotopy (co)fibre sequences in stable (infinity,1)-categories, subject to various axioms.
A central class of examples of triangulated categories are the derived categories $D(\mathcal{A})$ of abelian categories $\mathcal{A}$. These are the homotopy categories of the stable (∞,1)-categories of chain complexes in $\mathcal{A}$. However the notion also encompasses important examples coming from nonabelian contexts, like the stable homotopy category, which is the homotopy category of the stable (infinity,1)-category of spectra. Generally, it seems that all triangulated categories appearing in nature arise as homotopy categories of stable (infinity,1)-categories (though examples of “exotic” triangulated categories probably exist).
By construction, passing from a stable (infinity,1)-category to its homotopy category represents a serious loss of information. In practice, endowing the homotopy category with a triangulated structure is often sufficient for many purposes. However, as soon as one needs to remember the homotopy colimits and homotopy limits that existed in the stable (infinity,1)-category, a triangulated structure is not enough. For example, even the mapping cone in a triangulated category is not functorial. Hence it is often necessary to work with some enhanced notion of triangulated category, like stable derivators, pretriangulated dg-categories, stable model categories or stable (infinity,1)-categories. See enhanced triangulated category for more details.
Also I added a small history section:
The notion of triangulated category was developed by Jean-Louis Verdier in his 1963 thesis under Alexandre Grothendieck. His motivation was to axiomatize the structure existing on the derived category of an abelian category. Axioms similar to Verdier’s were given by Albrecht Dold and Dieter Puppe in a 1961 paper. A notable difference is that Dold-Puppe did not impose the octahedral axiom (TR4).
Please feel free to correct and improve.
I edited enhanced triangulated category. I also created a stub at pretriangulated A-infinity-category.
I have added statement and proof of the long exact sequences induced by a distinguished triangle, here.
I discovered a nice note by Andrew Hubery (here) in which a whole bunch of different formulations of the octahedral axiom are proven to be equivalent. One of them (“axiom B” in the note) manifestly axiomatizes just the existence of homotopy pushouts. That is really what one uses, explicitly or implicitly, when proving the octahedron from a stable model category: homotopy pushouts and their pasting law.
I have added pointer to Hubery’s note to the entry (here).
Octahedral axiom has higher hypersimplicial analogues; if they are satisfied some say that one has a strongly triangulated category, as studied e.g. by Lyubashenko. Now how the homotopy pushout approach could add those (the strongly triangulated are somewhat more natural than triangulated, some think).
I suppose the higher analogs are equivalent to the existence of homotopy colimits of more general shape than just homotopy pushouts.
That’s really what a triangulated category structure is: instead of a full stable model structure remembering all homotopy colimits, it axiomatizes only the existence of a few of them (homotopy pushouts, their masting law and the induced homotopy cofiber sequences).
In the end the real thing is stable model categories and stable $\infty$-categories.
added publication data for
added pointer to:
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