Standard $t$-structures
Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier
TL;DR
The authors develop a general framework for inducing t-structures on tensor products of presentable ∞-categories by combining a presentable ∞-category $\\mathcal{X}$ with a presentable stable ∞-category $\\mathcal{E}$ carrying an accessible t-structure. They construct a canonical t-structure on the tensor product $\\mathcal{X} \otimes \\mathcal{E}$ with coconnective part $\\mathcal{X} \otimes \\mathcal{E}_{\le 0}$ and connective part generated by $X \otimes E$ with $E \in \\mathcal{E}_{\ge 0}$; when $\\mathcal{X}$ is an ∞-topos, the connective part is explicitly $\\mathcal{X} \otimes \\mathcal{E}_{\ge 0}$ and the construction is compatible with filtered colimits and right completeness. A key unstable statement for ∞-topoi shows an equivalence between truncated pointed objects under the tensor product, and the full proof reduces to the spectra case, leveraging localization properties of tensoring and stable ∞-category technology. The work extends standard t-structures on sheaves to categories of $\\mathcal{E}$-valued sheaves and hypersheaves, providing a robust exactness framework for tensoring with ∞-topoi.
Abstract
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 $τ= (\mathcal{E}_{\geq 0}, \mathcal{E}_{\leq 0})$, 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.