Intermediate co-$t$-structures, two-term silting objects, $\tau$-tilting modules, and torsion classes

Abstract

If $\left(\mathsf{A},\mathsf{B}\right)$ and $\left({\mathsf{A}}^{\prime },{\mathsf{B}}^{\prime }\right)$ are co-$t$-structures of a triangulated category, then $\left({\mathsf{A}}^{\prime },{\mathsf{B}}^{\prime }\right)$ is called intermediate if $\mathsf{A}\subseteq {\mathsf{A}}^{\prime }\subseteq \Sigma \mathsf{A}$. Our main results show that intermediate co-$t$-structures are in bijection with two-term silting subcategories, and also with support $\tau$-tilting subcategories under some assumptions. We also show that support $\tau$-tilting subcategories are in bijection with certain finitely generated torsion classes. These results generalise work by Adachi, Iyama, and Reiten.