Extriangulated Category Research Articles

A characterization of extriangulated categories with triangulated structure

We give a characterization of the extriangulated categories which admit the structure of a triangulated category. We show that these are the extriangulated categories where for every object X in the extriangulated category, the morphism 0 → X is a deflation and the morphism X → 0 is an inflation. We thereby give an explicit proof to the statement that for a Frobenius extriangulated category ∐ where proj ( ∐ ) = 0 , its stable category is triangulated.

Gorenstein derived functors for extriangulated categories

Let ( C , E , s ) be an extriangulated category with a proper class ξ of E -triangles. In this paper, we study Gorenstein derived functors for extriangulated categories. More precisely, we first introduce the notion of the proper ξ ‐ G projective resolution for an object in C and define the functors ξ xt G P ( ξ ) and ξ xt G I ( ξ ) . Under some assumptions, we give some equivalent characterizations for ξ ‐ G projective and ξ ‐ G injective dimensions of objects in C by vanishing of such functors. As an application, our main results generalize Ren and Liu’s work. Moreover, our proof is not far from the usual module categories or triangulated categories.

Semibricks and the Jordan–Hölder property

Let [Formula: see text] be an extriangulated category and [Formula: see text] be a semibrick in [Formula: see text]. Let [Formula: see text] be the filtration subcategory generated by [Formula: see text]. We introduce the weak Jordan–Hölder property (WJHP) and Jordan–Hölder property (JHP) in [Formula: see text] and show that [Formula: see text] satisfies WJHP. Furthermore, [Formula: see text] satisfies JHP if and only if [Formula: see text] is proper.

Auslander–Reiten theory in extriangulated categories

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category C _ \underline {\mathscr {C}} of an extriangulated category C \mathscr {C} is a τ \tau -category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if C \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on C _ \underline {\mathscr {C}} , including Igusa–Todorov’s Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of C _ \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of C _ \underline {\mathscr {C}} . Finally we prove that any locally finite symmetrizable τ \tau -quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms.

Balanced pairs and recollements in extriangulated categories with negative first extensions

A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this paper. We prove that there exists a bijective correspondence between balanced pairs and proper classes [Formula: see text] with enough [Formula: see text]-projectives and enough [Formula: see text]-injectives. This can be regarded as a simultaneous generalization of corresponding results of Fu–Hu–Zhang–Zhu for triangulated categories and of Wand–Li-Huang for abelian categories. Besides, we show that if [Formula: see text] is a recollement of extriangulated categories, then balanced pairs in [Formula: see text] can induce balanced pairs in [Formula: see text] and [Formula: see text] under natural assumptions. As an application, this result generalizes a result by Fu–Hu–Yao in abelian categories. Moreover, it highlights a new phenomena when it is applied to triangulated categories.

Abelian Hearts of Twin Cotorsion Pairs on Extriangulated Categories

It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian. In this article, we consider a special class of hearts of twin cotorsion pairs induced by [Formula: see text]-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we can also see that such hearts are hereditary. As an application, this generalizes the work by Liu in the exact case, thereby providing new insights into the triangulated case.

Balanced Pairs on Triangulated Categories

Let [Formula: see text] be a triangulated category. We first introduce the notion of balanced pairs in [Formula: see text], and then establish the bijective correspondence between balanced pairs and proper classes [Formula: see text] with enough [Formula: see text]-projectives and [Formula: see text]-injectives. Assume that [Formula: see text] is the proper class induced by a balanced pair [Formula: see text]. We prove that [Formula: see text] is an extriangulated category. Moreover, it is proved that [Formula: see text] is a triangulated category if and only if [Formula: see text], and that [Formula: see text] is an exact category if and only if [Formula: see text]. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.

Rigid subcategories and related subcategories in extriangulated categories

In this paper, we firstly obtain the analogue of a Bongartz completion for functorially finite rigid subcategories in an extriangulated category and then we investigate the properties of n-cluster tilting subcategories in an extriangulated category for any positive integer n and show some equivalent characterization of them. Particularly, when the extriangulated category is 2-Calabi-Yau, we show any n-cluster tilting subcategories are in fact cluster tilting for any integer and they coincide with strongly contravariantly finite maximal n-rigid subcategories.

Homotopy cartesian squares in extriangulated categories

Abstract Let ( C , E , s ) \left({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}}) be an extriangulated category. Given a composition of two commutative squares in C {\mathcal{C}} , if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by Mac Lane [Categories for the Working Mathematician, Second edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998] for abelian categories and by Christensen and Frankland [On good morphisms of exact triangles, J. Pure Appl. Algebra 226 (2022), no. 3, 106846] for triangulated categories.

Relative cluster tilting subcategories in an extriangulated category

<abstract><p>Let $ \mathscr{B} $ be an extriangulated category which admits a cluster tilting subcategory $ \mathcal{T} $. We firstly introduce notions of $ \mathcal{T} $-cluster tilting subcategories and related subcategories. Then we prove there is a correspondence between $ \mathcal{T} $-cluster tilting subcategories of $ \mathscr{B} $ and support $ \tau $-tilting pairs of $ mod \underline{\Omega(\mathcal{T}}) $, which recovers several main results from the literature. Note that the generalization is nontrivial and we give a new proof technique.</p></abstract>

Consistent pairs of $ \mathfrak{s} $-torsion pairs in extriangulated categories with negative first extensions

<abstract><p>As a generalization of a consistent pair of $ t $-structures on triangulated categories, we introduced the notion of a consistent pair of $ \mathfrak{s} $-torsion pairs in the extriangulated setup. Let $ (\mathcal{T}_{i}, \mathcal{F}_{i}) $ be an $ \mathfrak{s} $-torsion pair in an extriangulated category with a negative first extension for any $ i = 1, 2 $. By using the consistent pair, we gave a criterion for $ (\mathcal{T}_{1}\ast\mathcal{T}_{2}, \mathcal{F}_{1}\cap\mathcal{F}_{2}) $ to be an $ \mathfrak{s} $-torsion pair. Our results were then applied to the torsion theory induced by $ \tau $-rigid modules.</p></abstract>

Relative extriangulated categories arising from half exact functors

Relative theories(=closed subfunctors) are considered in exact, triangulated and extriangulated categories by Dräxler-Reiten-Smalø-Solberg-Keller, Beligiannis and Herschend-Liu-Nakaoka, respectively. We give a construction method of closed subfunctors from given half exact functors which contains existing constructions. Moreover, if an extriangulated category has enough projective objects, then every closed subfunctor is obtained by this construction.

Intervals of s-torsion pairs in extriangulated categories with negative first extensions

AbstractAs a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$ -tilting reduction and brick labelling.

Localization of extriangulated categories

In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories by means of Hovey twin cotorsion pairs, and the localization of exact categories by two-sided admissibly percolating subcategories.

One-sided Frobenius pairs in extriangulated categories

Let be an extriangulated category with a proper class ξ of -triangles. We introduce the notions of left Frobenius pairs, left (n-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to ξ in We show how to construct left cotorsion pais from left n-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. We also study the relation between a certain class of cotorsion pairs and that of n-cotorsion pairs. These work generalize Ma-Zhao-Huang’s results in triangulated categories and partially generalize Becerril-Mendoza-Pérez-Santiago’s results in abelian categories.

𝒲(ξ)-Gorenstein objects in extriangulated categories

We fix a proper class of [Formula: see text]-triangles [Formula: see text] in an extriangulated category [Formula: see text]. Let [Formula: see text] be a class of objects in [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text] and all [Formula: see text]. In this paper, we study [Formula: see text]-Gorenstein objects, and use the relative homological methods to show that the stability of the Gorenstein category [Formula: see text] in [Formula: see text]. Moreover, we give some equivalent characterizations of [Formula: see text]-Gorenstein objects and finite [Formula: see text]-(co)resolution dimensions for any object in [Formula: see text]. As an application, we consider the equivalent characterizations of [Formula: see text]-[Formula: see text]projective objects, [Formula: see text]-[Formula: see text]projective dimension and their duals.

Relative resolution dimensions in extriangulated categories

Let $\mathscr{C}$ be an extriangulated category with enough projective and injective objects. We introduce the notions of relative re解 dimensions, relative core解 dimensions, $\mathcal{X}$-complete pairs and $\mathcal{X}$-hereditary pairs in this paper. Using $\mathcal{X}$-complete pair and $\mathcal{X}$-hereditary pair, we describe the relationship between homological dimensions and re解 dimensions of objects and subcategories respectively, and meanwhile we give some of the different descriptions for the realization of the heart of $\mathcal{X}$-complete pairs.

ξ -Tilting Objects in Extriangulated Categories

Extriangulated categories were introduced by Nakaoka and Palu via extracting the similarities between exact categories and triangulated categories. In this article we introduce the notion of [Formula: see text]-tilting objects in an extriangulated category, where [Formula: see text] is a proper class of [Formula: see text]-triangles. Our results extend the relative tilting theory in extriangulated categories.

Idempotent completion of extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As an application, we prove that the idempotent completion of a recollement of extriangulated categories is still a recollement.

A new method to construct model structures from left Frobenius pairs in extriangulated categories

<abstract><p>Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we first introduce the concept of left Frobenius pairs on an extriangulated category $ \mathcal{C} $, and then establish a bijective correspondence between left Frobenius pairs and certain cotorsion pairs in $ \mathcal{C} $. As an application, some new admissible model structures are established from left Frobenius pairs under certain conditions, which generalizes a result of Hu et al. (J. Algebra 551 (2020) 23–60).</p></abstract>