Purity in compactly generated derivators and t-structures with Grothendieck hearts

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 $\mathbf{t}=(\mathcal{U},\mathcal{V})$ on $\mathcal{T}$, we show that $\mathcal{V}$ is definable if and only if $\mathbf{t}$ is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to $\mathbf{t}$ being homotopically smashing and to $\mathbf{t}$ being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of $\mathbf{t}$ are determined by purity conditions on the associated partial cosilting object.

25 pages. The following changes were made to v2: Corrected statement of third theorem in the introduction; changed acknowledgement to include thanks to Michal Hrbek; corrected statement and proof of Proposition 5.6; added counter-example of original statement in Example 5.7; added Example 5.16, which shows that Theorem 5.14 does not hold for partial cosilting objects in general