In this article we provide a simple combinatorial description of morphisms between indecomposable complexes in the bounded derived category of a gentle algebra.

We study t-structures with Grothendieck hearts on compactly generated triangulated categories 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 T in terms of directed homotopy colimits. For a left nondegenerate t-structure t = (U, V) on T , we show that V is definable if and only if t is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to t being homotopically smashing and to t being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of t are determined by purity conditions on the associated partial cosilting object.

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Let Λ be a derived-discrete algebra. We show that the Krull-Gabriel dimension of the homotopy category of projective Λ-modules, and therefore the Cantor-Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobiński and Krause [9]. We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han [16]. Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective Λ-modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple.

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