Vol. 306, No. 2, 2020

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Definability and approximations in triangulated categories

Rosanna Laking and Jorge Vitória

Vol. 306 (2020), No. 2, 557–586
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.

Keywords
precover, preenvelope, definable subcategory, torsion pair, t-structure, recollement
Mathematical Subject Classification 2010
Primary: 18C35, 18E30, 18E35, 18E40
Milestones
Received: 30 April 2019
Revised: 16 December 2019
Accepted: 9 January 2020
Published: 13 July 2020
Authors
Rosanna Laking
Dipartimento di Informatica
Università degli Studi di Verona
Verona
Italy
Jorge Vitória
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari
Cagliari
Italy