# Sources for exact triangles in triangulated categories.

The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (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?

I would also appreciate answers focussing on triangulated categories in general.

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.
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).