Definability and approximations in triangulated categories

Abstract

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.

arXiv commentUpdated version with new Proposition 6.11

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Cited by in corpus (14)

• Silting objects
• $t$-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
• A functorial approach to rank functions on triangulated categories
• Duality pairs, phantom maps, and definability in triangulated categories
• Model theory in compactly generated (tensor-)triangulated categories
• Mutation and torsion pairs
• The shift-homological spectrum and parametrising kernels of rank functions
• Right triangulated categories: As extriangulated categories, aisles and co-aisles
• Gluing compactly generated t-structures over stalks of affine schemes
• Definable functors between triangulated categories
• Recollements induced by good silting objects
• Telescope conjecture via homological residue fields with applications to schemes