# What fails in constructing a homotopy category out of candidate triangles in a triangulated category?

Following Neeman’s article “New axioms for triangulated categories”, for a triangulated category $$mathscr T$$ let $$CT(mathscr T)$$ denote the category of candidate triangles, i.e. diagrams
$$begin{equation}Xoverset fto Yoverset gto Zoverset hto Sigma Xquad (*)end{equation}$$
such that $$gf=0$$, $$hg=0$$ and $$(Sigma f)h=0$$, with morphisms being commutative diagrams between such triangles.
We can define homotopy of maps between candidate triangles to be chain homotopy and there is an automorphism $$tildeSigmacolon CT(mathscr T)to CT(mathscr T)$$ which takes $$(*)$$ to
$$Yoverset{-g}to Zoverset{-h}to Sigma Xoverset{-Sigma f}to Sigma Y.$$
We can define mapping cones as in a usual chain complex category, and a lot of the usual results hold for this category (e.g. homotopic maps have isomorphic mapping cones).

What I fail to see, is why the mapping cone construction along with $$tildeSigma$$ does not give rise to a triangulation of $$CT(mathscr T)$$?