$t$-Structures on stable derivators and Grothendieck hearts

Abstract

We prove that given any strong, stable derivator and a $t$-structure on its base triangulated category $\mathcal{D}$, the $t$-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one. We use this to show that the $t$-structure being compactly generated implies that the coaisle is closed under directed homotopy colimit which in turns implies that the heart is an (Ab.$5$) Abelian category. If, moreover, $\mathcal{D}$ is a well generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) $t$-structure has a generator. As a consequence, it follows that the heart of any compactly generated $t$-structure of a well generated algebraic or topological triangulated category is a Grothendieck category.

62 pages

References in corpus (7)

• Higher Topos Theory
• Torsion pairs in silting theory
• Cotilting Sheaves on Noetherian Schemes
• The linearity of traces in monoidal categories and bicategories
• Two results from Morita theory of stable model categories
• On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functors
• The t-structure induced by an n-tilting module

Cited by in corpus (10)

• Silting objects
• Definability and approximations in triangulated categories
• $t$-Structures with Grothendieck hearts via functor categories
• Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings
• Hearts for commutative noetherian rings: torsion pairs and derived equivalences
• Tilting complexes and codimension functions over commutative noetherian rings
• Mutation and torsion pairs
• Gluing compactly generated t-structures over stalks of affine schemes
• Singular equivalences to locally coherent hearts of commutative noetherian rings
• Mutation and the Gabriel spectrum