ScalingStacks

We next show condition $(3)$. Suppose $X$ is an object of $𝒯$. By Proposition 3.6 there is an $S^{{⟂}_{\mathrm{\infty }}}$-preenvelope $\mu :X\to \overline{X}$. Write $B=\overline{X}$ and extend the morphism $\mu :X\to B$ to a distinguished triangle:

(4.1)

${\mathrm{\Sigma }}^{-1}A\to X\to B\to A.$

We claim that $\text{Hom}(S,{\mathrm{\Sigma }}^{i}A)=0$ for . Consider the following long exact sequence obtained from (4.1):

$$\text{Hom}(S,{\mathrm{\Sigma }}^{i-1}A)\to \text{Hom}(S,{\mathrm{\Sigma }}^{i}X)\to \text{Hom}(S,{\mathrm{\Sigma }}^{i}B)\to \text{Hom}(S,{\mathrm{\Sigma }}^{i}A).$$

Now, by Lemma 4.3, we see that $\text{Hom}(S,{\mathrm{\Sigma }}^{i}X)\to \text{Hom}(S,{\mathrm{\Sigma }}^{i}B)$ is an isomorphism for all . Hence $\text{Hom}(S,{\mathrm{\Sigma }}^{i}A)=0$ for all and $A\in 𝒜$. Hence the distinguished triangle in (4.1) above gives us the required distinguished triangle.