Question

The other day I came across the statement that in the triangulated category kk (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides extensions". Except for mapping cone triangles I don't know what is meant. What can you come up with?

Answer 1

This question has kind of been bothering me since I started thinking about it - I am far from an expert on KK-theory but I thought I'd throw something out there and maybe someone else will see it and come along and agree with me or correct me.

I think this statement is a way of thinking rather than something precise. Indeed by definition every distinguished triangle in KK is an isomorph of an extension triangle (although in the equivariant case I believe life is not so simple). Alternatively one can define the triangulation by taking distinguished triangles to be those candidates triangle (i.e. $X\to Y\to Z \to \Sigma X$ where each pair of composites vanishes) which are isomorphic to mapping cone triangles. So this is really all one has. The same story is true in the derived category of an abelian category for instance.

But (here is the punchline) one does not generally build the triangles one needs to prove things by considering short exact sequences of chain complexes! Indeed, one of the virtues of triangulated categories is that one has the ability to produce lots of new objects and triangles starting with very little. Often it is hard/impossible to do this in any explicit way - in fact one generally just knows that some collection of triangles doing the job exists and has no idea what they look like. So even though every triangle one might construct is (up to KK equivalence) an extension there is a very good chance that one didn't obtain it by writing down an explicit extension.

I guess there is also the fact that a triangle which is just isomorphic in KK to an extension triangle is not itself literally an extension of $C^*$-algebras. I don't know the stuff well enough to know whether or not one can produce interesting triangles via other constructions where there is a guarantee that some extension exists to make it distinguished. This is entirely possible (and in my opinion viewing such a construction as a different source of triangles is a worthwhile psychological distinction).

Answer 2

I have something to add: The octahedron axiom connects the mapping cones of two morphisms $f\colon A\to B$ and $g\colon B\to C$ with the mapping cone of $g\circ f$ by an exact triangle.