ScalingStacks

Proposition 2.4. Let $\mathcal{C}$ be an idempotent closed triangulated category and $\mathcal{T}$ be a collection of objects in $\mathcal{C}.$ Assume that $\mathcal{T}$ is negative and generates $\mathcal{C}$ with respect to isomorphisms, direct summands and triangles

$$\mathcal{C}=\langle\mathcal{T}\rangle_{\cong,\inplus,\Delta}.$$

Then there is a bounded weight structure on $\mathcal{C}$ whose heart

$$\mathcal{C}^{w=0}=\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus}$$

is the full subcategory of $\mathcal{C}$ generated by $\mathcal{T}$ under isomorphisms, direct summands and finite direct sums.