Morita theory for stable derivators

Abstract

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 $\mathbf{t}=({\mathcal{D}}^{\le 0},{\mathcal{D}}^{\ge 0})$ on the triangulated category $\mathbb{D}(1)$, and letting $\mathcal{A}={\mathcal{D}}^{\le 0}\cap {\mathcal{D}}^{\ge 0}$ 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.

62 pages