ScalingStacks

Definition 1.2. Let $S$ be an object of a triangulated category $𝒯$. The subcategory right $n$-perpendicular to $S$, denoted by $S^{{⟂}_n}$, is given by:

$$S^{{⟂}_n}:=\{X\in 𝒯|\text{Hom}(S,{\mathrm{\Sigma }}^{i}X)=0\text{ for }i=1,\mathrm{\dots },n\}.$$

The subcategory right $\mathrm{\infty }$-perpendicular to $S$, denoted by $S^{{⟂}_{\mathrm{\infty }}}$, is given by:

$$S^{{⟂}_{\mathrm{\infty }}}:=\{X\in 𝒯|\text{Hom}(S,{\mathrm{\Sigma }}^{i}X)=0\text{ for }i>0\}.$$

Similarly, one can also define the subcategories left $n$-perpendicular and left $\mathrm{\infty }$-perpendicular to $S$.