The $Q$-shaped derived category of a ring -- compact and perfect objects

In a previous work we constructed the $Q$-shaped derived category of any ring $A$ for any suitably nice category $Q$. The $Q$-shaped derived category of $A$, which is denoted by ${\mathcal{D}}_Q(A)$, is a generalization of the ordinary derived category. In this paper we prove that the $Q$-shaped derived category of $A$ is a compactly generated triangulated category. We also define perfect objects in ${\mathcal{D}}_Q(A)$ and prove that these constitute a triangulated subcategory, ${\mathcal{D}}_Q^{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$, of the category ${\mathcal{D}}_Q(A{)}^{\mathrm{c}}$ of compact objects in the $Q$-shaped derived category. The subcategories ${\mathcal{D}}_Q^{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ and ${\mathcal{D}}_Q(A{)}^{\mathrm{c}}$ coincide if and only if the former is thick.