1. Introduction

Since their introduction, triangulated categories have become a fundamental object of study in algebra, geometry and topology, where they encode all the homological and homotopic information of the objects under investigation. Their ubiquity is seen through examples such as the derived category of modules over a ring, the derived category of (quasi)coherent sheaves on a scheme, and the homotopy category of spectra, amongst a plethora of others. In fact, many examples of interest (including those recalled above) are instances of compactly generated triangulated categories - ones in which there is a set of ‘small’ objects, called compact objects, which generate the entire category through the triangulated structure.

The property of being compactly generated endows the triangulated category with many pleasant properties. One such property is the existence of a universal functor to a Grothendieck abelian category which sends triangles to long exact sequences, and preserves coproducts. This Grothendieck abelian category is actually nothing other than the category of additive contravariant functors from the compact objects to abelian groups, and the associated functor is just the (restricted) Yoneda embedding. If one is only interested in the triangles which get sent to short exact sequences under this functor, rather than long exact sequences, one picks out the pure triangles, and, using these, one can consider the pure structure of a compactly generated triangulated category.

Purity has long been a staple for understanding and describing the structure of Grothendieck abelian, and particularly module, categories, as well as a vital device in answering many representation theoretic questions, such as classification problems for finite dimensional algebras and (co)tilting theory. A wide range of such applications, along with further references, may be found in [Reference Göbel and Trlifaj25, Reference Jensen and Lenzing33, Reference Prest51] for instance. The ability to transfer the utility of purity from the abelian functor category to the compactly generated triangulated setting means purity has emerged as a useful tool in many approaches to understanding the structure of compactly generated triangulated categories. For instance, several fundamental questions in homological algebra, representation theory, and homotopy theory can be considered through the lens of purity in such triangulated categories. We provide some examples as motivation for our study.

The telescope conjecture asks whether the kernel of every smashing localization is generated by a set of compact objects. Krause [Reference Krause37] used purity as a central tool to reframe the conjecture in an algebraic form and then prove a modified version of it. This conjecture was recently disproved for the stable homotopy category of spectra [Reference Burklund, Hahn, Levy and Schlank17], but it remains interesting to study the conjecture in other compactly generated triangulated categories. Margolis's uniqueness conjecture for the stable homotopy category is another significant open problem. It asks whether the finite spectra determine the whole stable homotopy category, that is, whenever $\mathsf {T}$ is a compactly generated triangulated category with compact objects $\mathsf {T}^{\mathrm {c}}$ such that $\mathsf {T}^{\mathrm {c}}$ is equivalent to the category of finite spectra, then $\mathsf {T}$ is equivalent to the category of spectra. It was shown in [Reference Christensen and Strickland18] that this conjecture can be reduced, through the lens of purity, to a more elementary equivalent question.

On the more algebraic side, since their introduction in [Reference Beĭlinson, Bernstein and Deligne15] where they were used to define perverse sheaves, $t$-structures have become a fundamental tool in understanding derived categories of sheaves and modules. The existence of $t$-structures is dependent on the existence of certain approximations or adjoints. The existence of approximations by a class of objects can in turn be determined through checking whether that class is closed under certain properties, often pure subobjects or pure quotients [Reference Crivei, Prest and Torrecillas20, Reference Krause40]. We will return to this as an application of the framework we develop later on, see theorem 4.3. In representation theory, $t$-structures and torsion pairs play a key role in the study of silting objects [Reference Angeleri Hügel, Marks and Vitória4, Reference Marks and Vitória43], which are a generalization of tilting objects as considered by Rickard [Reference Rickard54]. There are different motivations for studying silting objects: firstly, they are more flexible than tilting objects when it comes to mutation - one can obtain a new silting object from an old one [Reference Hügel, Laking, Šťovíček and Vitória32]. Secondly, silting objects can be seen as a generalization of injective modules to triangulated categories. We will consider this perspective in the main body of the paper.

One fundamental property of purity in module categories, and more generally Grothendieck abelian categories, is that a pure exact sequence is, in some sense, a weak version of a split exact sequence: a sequence is pure exact if and only if its image under the character dual $\mathrm {Hom}_{\mathbb {Z}}(-,\,\mathbb {Q}/\mathbb {Z})$ splits. However, unlike checking if a class is closed under direct summands, checking if a class has desirable purity closure properties is usually a difficult problem. For categories of modules over a ring, a useful concept, called a duality pair, was introduced by Holm and Jørgensen in [Reference Holm and Jørgensen30] to determine pure closure properties. Such a pair consists of two classes of modules $(\mathsf {A},\,\mathsf {B})$ such that $M \in \mathsf {A}$ if and only if $\mathrm {Hom}_\mathbb {Z}(M,\,\mathbb {Q}/\mathbb {Z}) \in \mathsf {B}$, and $\mathsf {B}$ is closed under summands and finite direct sums. They used this splitting property of pure exact sequences to prove that if $(\mathsf {A},\,\mathsf {B})$ is a duality pair then $\mathsf {A}$ has good pure closure properties, and frequently yields approximations.

In this paper we provide an axiomatic framework for detecting pure triangles in compactly generated triangulated categories through an exact functor that generalizes the character dual functor from the Grothendieck abelian setting. This enables us to define duality pairs, thus providing a scaffold from which one can verify pure closure properties. Moreover, we illustrate how duality pairs can, as suggested by the prior discussion, be used to determine the existence of approximations.

These results are widely applicable: all algebraic triangulated categories, as well as all rigidly-compactly generated tensor triangulated categories, fall within the framework. Following this abstract setup, we provide many examples which arise naturally in algebra and topology, before turning our attention to applications, which predominantly lie in representation theory and stable homotopy theory. We also consider the relationship between the duality pairs we define, and those defined in [Reference Holm and Jørgensen30], as well as the well established relationship between a compactly generated triangulated category, and its functor category.

A further motivation for us to develop the framework enabling us to construct duality pairs in the triangulated setting is the goal of better understanding definable subcategories. These are the subcategories which are determined by the vanishing of coherent functors, or equivalently in the presence of an enhancement, by closure under pure subobjects, products, and directed homotopy colimits. We emphasize that definable subcategories need not be triangulated.

Definable subcategories have long played an important role in representation theory, and understanding module categories [Reference Prest51, Reference Prest52]. However, it has been shown that they are of equal utility in the triangulated setting: for example, from an algebraic standpoint, every silting class is definable, as are Auslander and Bass classes of complexes, and many other classes of complexes of finite (co)homological dimension. Further motivation from the topological viewpoint is Freyd's generating hypothesis, which conjectures that a map $f$ between finite spectra is null homotopic if and only if the induced map on homotopy $\pi _*(f)$ is zero. This can be formulated in terms of definability [Reference Krause38], and as such the techniques of purity may provide new insight, just as in the seminal work of Krause on the telescope conjecture [Reference Krause37].

We now discuss the contents and main results of this paper in more detail. In order to construct an axiomatic setting in which we can consider duality pairs, we first need a functor abstracting the character dual with the aim of connecting pure and split triangles as in the module case. By distilling the salient features of the character dual, we define a duality triple $(\mathsf {T},\, \mathsf {U},\, Q)$; see definition 3.4. This consists of a pair of compactly generated triangulated categories $\mathsf {T}$ and $\mathsf {U}$ and exact functors $Q\colon \mathsf {T}^\text {op} \to \mathsf {U}$ and $Q\colon \mathsf {U}^\text {op} \to \mathsf {T}$ satisfying various natural properties which mirror those in the Grothendieck abelian setting. Our first main result shows that the structure of a duality triple completely determines the pure structure on a triangulated category.

We show that almost all compactly generated triangulated categories that arise in algebraic, geometric, and topological settings admit the structure of a duality triple. Thus the previous theorem is widely applicable: we prove in proposition 3.13 that any rigidly-compactly generated tensor-triangulated category $\mathsf {T}$ has the natural structure of a duality triple $(\mathsf {T},\,\mathsf {T},\,\mathbb {I})$ where $\mathbb {I}$ is the Brown–Comenetz dual. We also show that any compactly generated algebraic triangulated category has the natural structure of a duality triple via a functor induced by the character dual, see proposition 3.14. Therefore the above theorem has applications to a broad range of categories, such as the homotopy category of spectra, the derived category of a ring, quasicoherent sheaves on a scheme, stable categories of Frobenius categories, and certain categories of motives.

If one is given a duality triple $(\mathsf {T},\, \mathsf {U},\, Q)$ one may define a duality pair $(\mathsf {A},\,\mathsf {B})$, where $\mathsf {A}\subset \mathsf {T}$ and $\mathsf {B}\subset \mathsf {U}$ in the same way as in [Reference Holm and Jørgensen30]: the pair $(\mathsf {A},\,\mathsf {B})$ is a duality pair provided $\mathsf {B}$ is closed under summands and finite direct sums, and $A\in \mathsf {A}$ if and only if $Q(A)\in \mathsf {B}$. Using this definition, we are able to apply theorem 3.6 to show that, as in the case for modules over a ring, duality pairs in the triangulated setting also admit powerful pure closure properties.

We combine the preceding theorem with results from [Reference Laking and Vitória42], to obtain criteria for when $\mathsf {A}$ provides approximations, or appears as a torsion class. Consequently, as claimed, duality pairs can be a useful tool in understanding inclusions of subcategories and the existence of adjoints.

As mentioned above, one of our motivations for studying duality pairs is to provide a better understanding of definable subcategories, and we now turn our attention to this problem. Definable subcategories are determined by finitely presented functors, and therefore any interplay between duality pairs and definable categories must entail some way to put a duality on finitely presented functors. We provide such a duality by introducing the refined concept of an Auslander–Gruson–Jensen duality triple; despite this refinement, the main examples of duality triples we consider in this paper satisfy this additional requirement. We show in theorem 6.13 that this refined structure is precisely what is needed to establish an exact antiequivalence $\delta \colon (\mathsf {T}^{\mathrm {c}},\,\mathbf {Ab})^{\text {fp}}\to (\mathsf {U}^{\mathrm {c}},\,\mathbf {Ab})^{\text {fp}}$ between the categories of finitely presented additive functors on $\mathsf {T}^{\mathrm {c}}$ and $\mathsf {U}^{\mathrm {c}}$. Using $\delta$, and the bijective correspondence between definable subcategories of $\mathsf {T}$ and Serre subcategories of $(\mathsf {T}^{\mathrm {c}},\,\mathbf {Ab})^\mathrm{fp}$, we define the dual definable subcategory, denoted $\mathcal {D}^{d}\subset \mathsf {U}$, of a definable subcategory $\mathcal {D}$ of $\mathsf {T}$. One would at this point hope that the dual definable subcategory should be related to duality pairs; the following theorem shows that this is indeed the case, and moreover that definable classes are completely characterized via duality pairs.

The above theorem is also shown to hold in the setting of Grothendieck abelian categories, where it is also new. Both results completely characterize definable subcategories in terms of symmetric duality pairs and closure under a single property - either coproducts or products. Although the above theorem makes no mention of triangulated subcategories, we obtain the following corollaries: firstly, a definable category $\mathcal {D}$ is closed under triangles if and only if its dual definable category is (corollary 6.19), and secondly, if $\mathsf {T}$ has a compatible tensor product, then $\mathcal {D}$ is a $\otimes$-ideal if and only if its dual definable category is (proposition 8.1). We note that these corollaries were previously observed in [Reference Wagstaffe60] using completely independent machinery, but our techniques provide substantially more elementary and brief proofs.

Having described the main abstract results and tools, we turn to explaining the applications we obtain. The first of these is in representation theory, where we consider the relationship between duality pairs and silting theory, see § 7. The main result of this section is the following:

This result is a generalization of a well established result about the duality between tilting and cotilting modules, and their induced classes, in module categories. It is well known that silting and cosilting class are definable [Reference Marks and Vitória43], and as such the contribution of the previous result is twofold. Firstly, it shows that the character dual on the derived category passes to a duality on silting and cosilting classes recovering a result of [Reference Angeleri Hügel and Hrbek3], and secondly, that moreover, this duality corresponds to the duality on definable subcategories.

Our second major application is in the study of tensor-triangulated categories. Many triangulated categories in nature have the additional structure of a compatible monoidal structure, and this monoidal structure often opens the door to new approaches of study. For example, a scheme cannot be reconstructed from the triangulated structure on its derived category alone, but can be reconstructed from the tensor-triangular structure [Reference Balmer7]. As such, the study of tensor-triangulated categories is a generalization of algebraic geometry, and algebro-geometric techniques and constructions have now become a vital part of the toolkit. For example, just as is standard in algebraic geometry, the global structure of a tensor-triangulated category can often be determined via performing localizations and then gluing together the results.

Therefore we often seek to understand the global structure of a category via the behaviour of certain types of subcategories. For example, there is a set of thick tensor ideals of compacts so we can understand the structure of the compact objects via this, but to understand big objects we have to go further. The natural extension of this to the big setting, is that of localizing tensor ideals, however, it is not known whether there is a set of these. On the other hand, there is only a set of definable subcategories, so they provide a way to extend the paradigm from small objects to big objects. Moreover, definable classes also encompass many subcategories of substantial interest: for example, the set of smashing subcategories is in bijection with the set of triangulated definable subcategories.

Just as the Zariski spectrum of a commutative ring $R$ provides a local-to-global approach for understanding the structure and properties of $R$-modules, there is a categorification of this to tensor-triangulated categories. In fact, associated to a tensor-triangulated category, there are many ‘spectra’ of interest, each isolating a particular feature. For example, the Balmer spectrum determines the collection of thick tensor ideals [Reference Balmer7], and the Ziegler spectrum determines the definable subcategories [Reference Bird and Williamson16, Reference Prest51]. In this vein, our main application to tensor-triangulated categories determines the interaction between localizations and definability, and hence implicitly between localization and the Ziegler spectrum.

As previously discussed, purity in a compactly generated triangulated category $\mathsf {T}$ is determined by purity in the category $\mathrm {Mod}(\mathsf {T}^{\mathrm {c}})$ of additive functors $(\mathsf {T}^{\mathrm {c}})^{\mathrm {op}}\to \mathbf {Ab}$. Since duality pairs concern purity, one may naturally ask whether duality pairs in $\mathsf {T}$ can be related to duality pairs in $\mathrm {Mod}(\mathsf {T}^{\mathrm {c}})$. This is the main study of § 9, where we show that by using a nerve construction one can construct an appropriate structure to define duality pairs on the corresponding functor categories. By considering an assignment $(-)_0$ which produces a class in $\mathsf {T}$ from a class in $\mathrm {Mod}(\mathsf {T}^{\mathrm {c}})$ by restricting to representable functors, we are able to show that all duality classes in an Auslander–Gruson–Jensen duality triple come from a duality class in a functor category. Moreover, this assignment refines to duality pairs that are symmetric, and even definable. The main results are summarized in the following theorem.

In particular, as an application, we show in example 9.7 that every duality pair in a category of modules can be recovered from a duality pair on the derived category. Consequently, the framework developed in this paper extends and completely subsumes the classical concept introduced in [Reference Holm and Jørgensen30].