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Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.

The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large commutative diagram is particularly quizzical. While I am ok with understanding these axioms I was hoping to ask two questions about them.

1) What was the classical motivation for these axioms? Was there a particular example in mind to conform to?

and

2) Is there a modern motivating example for these axioms that differs from the classical?

I understand these things much better when I have specific examples to keep in mind, and since I am learning these in a general context, right now that is lacking. I was hoping you all could fill me in.

Thanks in advance!

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    $\begingroup$ I imagine that Keller has an article in his webpage on this subject, but I imagine you already looked for it! Can you provide a link? Or maybe a reference? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 21, 2010 at 5:37
  • $\begingroup$ These are just analogue of axioms of Grothendieck pretopology which is explained in Rosenberg's paper.. $\endgroup$
    – Shizhuo Zhang
    Commented Feb 4, 2010 at 21:21
  • $\begingroup$ Keller did not introduce cosuspend category but suspend category. Cosuspend category is due to Rosenberg who considered the right exact structure instead of left exact structure.. $\endgroup$
    – Shizhuo Zhang
    Commented Feb 4, 2010 at 23:51
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    $\begingroup$ Shizhuo, I know you know Rosenberg has written more than one paper... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Mar 8, 2010 at 18:26

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If my memory is right, the stable category of a Quillen exact category is just Keller-suspended; under additional hypothesis, e.g. Frobenius it is in fact triangulated i.e. Keller-suspended with an invertible suspension (shift). This gives a lot of examples. Sasha Rosenberg's generalizations are in nonadditive setup.

B. Keller, Chain complexes and stable categories, Manus. Math. 67 (1990), 379-417.

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