ScalingStacks

Definition 1.4. Let $𝒯$ be a triangulated category. A pair of full subcategories of $𝒯$, $(𝒜,ℬ)$, will be called a co-$t$-structure on $𝒯$ if it satisfies the following properties:

$(0)$ $𝒜$ and $ℬ$ are closed under direct summands;

$(1)$ ${\mathrm{\Sigma }}^{-1}𝒜\subseteq 𝒜$ and $ℬ\subseteq {\mathrm{\Sigma }}^{-1}ℬ$;

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

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

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

with $A\in 𝒜$ and $B\in ℬ$.