\(t\)-structures on stable derivators and Grothendieck hearts (Q6175943)

The paper under review contains an extensively study of and gives positive answers to the following natural questions: Given a \(t\)-structure in the base triangulated category of strong stable derivator, is it possible to lift it canonically to all triangulated categories of coherent diagrams? Further, given an incoherent diagram in the heart of such a \(t\)-structure, is it possible to lift it, even uniquely, to a coherent one? In order to be more precise, let \(\mathcal D\) be a triangulated category with the shift functor denoted by \(\Sigma\). Recall that a \(t\)-structure in a triangulated category is a pair of subcategories \({\mathbf t}=(\mathcal U,\Sigma\mathcal V)\), such that \(\mathcal U\) is closed under positive shifts (automatically \(\mathcal V\) is closed under negative shifts), \(\mathcal D(U,V)=0\) for all \(U\in\mathcal U\) and all \(V\in\mathcal V\) and any objects \(X\in\mathcal D\) lies in a triangle \(U_X\to X\to V_X\to\Sigma U_X\), with \(U_X\in\mathcal U\) and \(V_X\in\mathcal V\). Moreover this triangle is functorial in \(X\) and gives rise to a right, respectively left adjoint for the inclusion functors \(\mathcal U\to \mathcal D\), and \(\mathcal V\to\mathcal D\). The heart of this \(t\)-structure is defined to be \(\mathcal H=\mathcal U\cap\mathcal V\) and it is known to be abelian. Suppose further that \(\mathcal D\) is the base category of a strong stable derivator, that is \(\mathcal D=\mathbb{D}(\mathbf 1)\), for \(\mathbb{D}:\mathbf{Cat}^{\mathrm{op}}\to\mathbf{CAT}\). Then Theorem A in the paper says that for every small category \(I\) we get a \(t\)-structure \({\mathbf t}_I=(\mathcal U_I,\Sigma\mathcal V_I)\) in \(\mathbb{D}(I)\), where \[ \mathcal U_I=\{X\in\mathbb{D}(I)\mid X_i\in\mathcal U, \forall i\in I\}, \] \[ \mathcal V_I=\{Y\in\mathbb{D}(I)\mid Y_i\in\mathcal V, \forall i\in I\}. \] Moreover the diagram functor induces an equivalence between the heart \(\mathcal H_I\) of \({\mathbf t}_I\) and \(\mathcal H^I\). The study of the abstract problem above is motivated by a more concrete question, namely when is the heart of a (nice enough) \(t\)-structure is a Grothendieck category. An application of the lifting property for \(t\)-structures shows that if \(\mathcal D=\mathbb{D}(\mathbf 1)\) is as above, and \({\mathbf t}=(\mathcal U,\Sigma\mathcal V)\) is compactly generated, then \(\mathcal V\) is closed under homotopy colimits and further \(\mathcal H\) has exact colimits (see Theorem B). If, in addition, \(\mathcal D\) is a well-generated algebraic or topological triangulated category, then the heart of any compactly generated \(t\)-structure has a generator, therefore it is a Grothendieck abelian category.