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

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## 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.

## Keywords

T-structure Cosilting Cotilting Purity Definable Reduced product Derivator Homotopically smashing Locally coherent Locally noetherian## Mathematics Subject Classification

18E15 18E30 03C20## Notes

### Acknowledgements

The author would like to thank Moritz Groth and Gustavo Jasso for many interesting and helpful conversations about derivators and other higher categorical structures. She would like to thank Lidia Angeleri Hügel, Frederik Marks and Jorge Vitória for discussions regarding the definition of partial cosilting, which also led to the contents of Example 4.4. Particular thanks are extended to Prof. Angeleri Hügel for her ongoing support of this project. During the later stages of this project, the author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 797281.

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