If you'll have patience with me, I understand that this is not the first time that a variation of this question is being asked on MathOverflow, but alas, I am unable to truly make sense of those answers, so pray have patience with me.
The problem I have is that I frankly find myself unable to truly understand how distinguished triangles (or exact triangles, depending on your taste) are generalizations of short exact sequences.
I suppose that what I would really need to get this through my skull would be to see an explanation that proceeds in one of two fashions:
- You start with a triangulated category, add some additional structure to make it into an abelian category, and show how that renders all distinguished triangles into short exact sequences.
- You start with an abelian category, and look at a generic short exact sequence of three objects. You remove some structure so as to make the category merely triangulated, and show how in doing so, while before you could say that the three objects were related by a short exact sequence, now you can only say that they are related by means of a distinguished triangle.
Most ideally, I'd like to see both explanations of course.
I thank you in advance. Such an explanation would make life far easier for me.