ScalingStacks

Definition 1.3. Let $𝒯$ be a triangulated category. A pair of full subcategories of $𝒯$, $(𝒳,𝒴)$, is called a $t$-structure on $𝒯$ if it satisfies the following properties:

$(1)$ $𝒳\subseteq {\mathrm{\Sigma }}^{-1}𝒳$ and ${\mathrm{\Sigma }}^{-1}𝒴\subseteq 𝒴$;

$(2)$ $\text{Hom}(𝒳,{\mathrm{\Sigma }}^{-1}𝒴)=0$;

$(3)$ For any object $Z$ of $𝒯$ there exists a distinguished triangle

$$X\to Z\to {\mathrm{\Sigma }}^{-1}Y\to \mathrm{\Sigma }X$$

with $X\in 𝒳$ and $Y\in 𝒴$.