MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


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!

share|cite|improve this question
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 '10 at 5:37
These are just analogue of axioms of Grothendieck pretopology which is explained in Rosenberg's paper.. – Shizhuo Zhang Feb 4 '10 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 '10 at 23:51
Shizhuo, I know you know Rosenberg has written more than one paper... – Mariano Suárez-Alvarez Mar 8 '10 at 18:26

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.