Standard $t$-structures

We provide a general construction of induced $t$-structures, that generalizes standard $t$-structures for $\mathrm{\infty }$-categories of sheaves. More precisely, given a presentable $\mathrm{\infty }$-category $\mathcal{X}$ and a presentable stable $\mathrm{\infty }$-category $\mathcal{E}$ equipped with an accessible $t$-structure $\tau =({\mathcal{E}}_{\ge 0},{\mathcal{E}}_{\le 0})$, we show that $\mathcal{X}\otimes \mathcal{E}$ is equipped with a canonical $t$-structure whose coconnective part is given in $\mathcal{X}\otimes {\mathcal{E}}_{\le 0}$. When $\mathcal{X}$ is an $\mathrm{\infty }$-topos, we give a more explicit description of the connective part as well.