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