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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!

flag	asked Jan 21 2010 at 4:51 B. Bischof 2,621●22●34

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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? – Mariano Suárez-Alvarez Jan 21 2010 at 5:37

These are just analogue of axioms of Grothendieck pretopology which is explained in Rosenberg's paper.. – Shizhuo Zhang Feb 4 2010 at 21:21

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.. – Shizhuo Zhang Feb 4 2010 at 23:51

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Shizhuo, I know you know Rosenberg has written more than one paper... – Mariano Suárez-Alvarez Mar 8 2010 at 18:26

Zoran Škoda · Answer 1 · 2010-03-08 16:37:03Z UTC

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.

Motivation for Cosuspended Category Axioms

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