ScalingStacks

Definition 2.1. Let $\mathcal{C}$ be a triangulated category. A weight structure $w$ on $\mathcal{C}$ is a pair $w=(\mathcal{C}^{w\leq 0},\mathcal{C}^{w\geq 0})$ of idempotent-closed full subcategories of $\mathcal{C},$ such that with $\mathcal{C}^{w\leq n}:=\mathcal{C}^{w\leq 0}[-n]$ and $\mathcal{C}^{w\geq n}:=\mathcal{C}^{w\geq 0}[-n]$ the following conditions are satisfied:

1. (1)

   $\mathcal{C}^{w\leq 0}\subseteq\mathcal{C}^{w\leq 1}$ and $\mathcal{C}^{w\geq 1}\subseteq\mathcal{C}^{w\geq 0};$

2. (2)

   for all $X\in\mathcal{C}^{w\geq 0}$ and $Y\in\mathcal{C}^{w\leq-1}$, we have $\operatorname{Hom}_{\mathcal{C}}(X,Y)=0;$

3. (3)

   for any $X\in\mathcal{C}$ there is a distinguished triangle

   with $A\in\mathcal{C}^{w\geq 1}$ and $B\in\mathcal{C}^{w\leq 0}.$

The full subcategory $\mathcal{C}^{w=0}=\mathcal{C}^{w\leq 0}\cap\mathcal{C}^{w\geq 0}$ is called the heart. A weight structure is called bounded if $\bigcup_{i}\mathcal{C}^{w\leq i}=\bigcup_{i}\mathcal{C}^{w\geq i}=\mathcal{C}.$ A triangulated functor $F:\mathcal{C}\to\mathcal{D}$ between two categories with weight structures is called weight exact if $F(\mathcal{C}^{w\leq 0})\subset\mathcal{D}^{w\leq 0}$ and $F(\mathcal{C}^{w\geq 0})\subset\mathcal{D}^{w\geq 0}.$