ScalingStacks

Definition 3.2. The category of mixed Tate motives²² 2 In the literature, there are many notations for the category of mixed Tate motives. For example, $\operatorname{TDM}$ in [HK06], $\operatorname{MTDer}$ in [SW18, SVW18], $\operatorname{DMT}$ in [Spi16] or $\operatorname{DTM}$ in [Lev05]. is the subcategory

$$\operatorname{DTM}(k,\mathbb{Q})=\langle\mathbb{Q}\rangle_{\cong,\inplus,\Delta,(\pm 1)}\subset\operatorname{DM}(k,\mathbb{Q})$$

generated by $\mathbb{Q}$ under isomorphism, direct summands, Tate twists and triangles. The category of pure Tate motives is the category generated by objects $\mathbb{Q}(n)[2n]$

$$\operatorname{DTM}(k,\mathbb{Q})^{w=0}=\langle\mathbb{Q}(n)[2n]\mid n\in\mathbb{Z}\rangle_{\cong,\inplus,\oplus}\subset\operatorname{DTM}(k,\mathbb{Q})$$

with respect to isomorphism, direct summands and finite direct sums.