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

TL;DR

This paper characterizes pure triangles and definable subcategories in compactly generated derivators, linking t-structures with Grothendieck hearts to purity, smashing conditions, and cosilting objects in triangulated categories.

Contribution

It provides an intrinsic characterization of purity and definability in t-structures with Grothendieck hearts, connecting these to homotopy colimits and cosilting objects.

Findings

Definable subcategories characterized by directed homotopy colimits.

Equivalence of smashing, homotopically smashing, and cosilting conditions.

Finiteness conditions on the heart relate to purity of cosilting objects.

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.