Localized cone triangles satisfy tr one through tr three

The additive localization exists. Since a quasi-isomorphism remains one after either shift, shifting both arrows of a roof defines mutually inverse additive shifts. The zero complex and identity triangles descend as well.

For any arrow write $\alpha =Q(s{)}^{-1}Q(f)$ with $s:Y\to Y^{\mathrm{\prime }}$ a quasi-isomorphism. A cone triangle on $f:X\to Y^{\mathrm{\prime }}$ descends and transport along $Q(s)$ completes $\alpha :X\to Y$. Closure under isomorphism is built into the definition. Rotation gives $(Y,Z,X[1],v,w,-u[1])$ because this is the rotation in $K$; shifting a roof preserves the minus sign. This proves TR1 and both directions of TR2.

For TR3 first replace the two triangles by images of triangles on ordinary maps $f,f^{\mathrm{\prime }}$. Apply the square-clearing lemma to the prescribed first two components. It supplies a third ordinary arrow $f^{\mathrm{\prime }\mathrm{\prime }}$ and two commuting squares with maps $(k,l)$ and denominator maps $(s,t)$ into $f^{\mathrm{\prime }\mathrm{\prime }}$. Complete each square to a morphism of cone triangles in $K$ by its TR3, giving third maps $m:C_f\to C_{f^{\mathrm{\prime }\mathrm{\prime }}}$ and $r:C_{f^{\mathrm{\prime }}}\to C_{f^{\mathrm{\prime }\mathrm{\prime }}}$.

For every $n$, take the five consecutive terms $H^n(X^{\mathrm{\prime }})$, $H^n(Y^{\mathrm{\prime }})$, $H^n(C_{f^{\mathrm{\prime }}})$, $H^{n+1}(X^{\mathrm{\prime }})$, $H^{n+1}(Y^{\mathrm{\prime }})$ and their counterparts for $f^{\mathrm{\prime }\mathrm{\prime }}$. Exactness holds in $K$; four vertical maps are isomorphisms because $s,t$ are quasi-isomorphisms. The five lemma makes $H^n(r)$ an isomorphism. Hence $r$ is a quasi-isomorphism before any triangulation of $D$ is used.

Put the third localized component equal to $Q(r{)}^{-1}Q(m)$. The two morphisms of triangles in $K$, with the now invertible comparison $Q(r)$, give all three commuting triangle squares and the shifted first component. Transporting back proves TR3 for the original data.