Morita theory for stable derivators

We give a general construction of realization functors for $t$-structures on the base of a strong stable derivator. In particular, given such a derivator $\mathbb D$, a $t$-structure $\mathbf t=(\mathcal D^{\leq0},\mathcal D^{\geq0})$ on the triangulated category $\mathbb D(\mathbb 1)$, and letting $\mathcal A=\mathcal D^{\leq0}\cap \mathcal D^{\geq0}$ be its heart, we construct, under mild assumptions, a morphism of prederivators $$\mathrm{real}_{\mathbf t}\colon \mathbf{D}_{\mathcal A}\to \mathbb D$$ where $\mathbf{D}_{\mathcal A}$ is the natural prederivator enhancing the derived category of $\mathcal A$. Furthermore, we give criteria for this morphism to be fully faithful and essentially surjective. If the $t$-structure $\mathbf t$ is induced by a suitably "bounded" co/tilting object, $\mathrm{real}_{\mathbf t}$ is an equivalence. Our construction unifies and extends most of the derived co/tilting equivalences appeared in the literature in the last years.