Purity in compactly generated derivators and t-structures with Grothendieck hearts
Rosanna Laking
TL;DR
The paper develops a purity framework for compactly generated triangulated categories that are underlying strong and stable derivators, characterising pure triangles and definable subcategories via coherent reduced products and directed homotopy colimits. It proves that for a left nondegenerate t-structure ${f t}=(\, ext{U},\V)$ with a Grothendieck heart, the following are equivalent: ${f t}$ is cosilting (partial) with a pure-injective object; ${\V}$ is definable; ${\bf t}$ is homotopically smashing; and ${\bf t}$ is smashing with a Grothendieck heart. The results unify approaches via purity (cosilting/cosilting objects) and homotopical smashing, and further relate purity to finiteness properties in the heart (locally noetherian or locally coherent) through elementary cogenerators and torsion theories of finite type. Overall, the work provides intrinsic criteria linking purity, t-structures with Grothendieck hearts, and finiteness conditions in a unified derivator framework, with concrete glued-t-structure constructions illustrating the theory.
Abstract
We study t-structures with Grothendieck hearts on compactly generated triangulated categories $\mathcal{T}$ that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of $\mathcal{T}$ in terms of directed homotopy colimits. For a left nondegenerate t-structure ${\bf t}=(\mathcal{U},\mathcal{V})$ on $\mathcal{T}$, we show that $\mathcal{V}$ is definable if and only if ${\bf t}$ is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to ${\bf t}$ being homotopically smashing and to ${\bf t}$ being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of ${\bf t}$ are determined by purity conditions on the associated partial cosilting object.