Duality pairs, phantom maps, and definability in triangulated categories

TL;DR

This paper introduces duality pairs and triples in triangulated categories, providing tools to analyze their properties, including a new characterization of phantom maps, and explores their applications in algebraic and homotopical contexts.

Contribution

It develops a framework for duality pairs and triples, introduces an axiomatic duality, and connects these concepts to definability and approximation in triangulated categories.

Findings

Characterization of phantom maps.

Equivalence of dual definable categories and symmetric coproduct closed duality pairs.

Applications to silting theory and stratified tensor-triangulated categories.

Abstract

We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure quotients and pure extensions, as well as providing a way to show the existence of approximations. One key ingredient is a new characterisation of phantom maps. We then introduce an axiomatic form of Auslander-Gruson-Jensen duality, from which we define dual definable categories, and show that these coincide with symmetric coproduct closed duality pairs. This framework is ubiquitous, encompassing both algebraic triangulated categories and stable homotopy theories. Accordingly, we provide many applications in both settings, with a particular emphasis on silting theory and stratified tensor-triangulated categories.