The Karoubi envelope and weak idempotent completion of an extriangulated category

Msapato, Dixy

doi:10.1007/s10485-021-09664-8

Cited by 9 publications

(8 citation statements)

References 16 publications

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“…In this section, we will construct a biadditive functor F : C op × C → Ab (see Subsection 4.1) and an exact realisation t of F (see Subsection 4.2), and then show that ( C, F, t) is an nexangulated category (see Subsections 4.3-4.5). For n = 1, we recover the main results of [Msa22]. First, we establish some notation to help our exposition.…”

Section: The Idempotent Completion Of An N-exangulated Categorymentioning

confidence: 86%

“…(iii) extriangulated (see [Msa22,Thm. 3.1]); and (iv) (n + 2)-angulated, where n 1 is an integer (see Lin [Lin21,Thm. 3.1]).…”

Section: Introductionmentioning

confidence: 99%

“…1.5] to equip C with a triangulation ( Σ, T ). Or, instead, one could view C as an extriangulated category and apply case (iii), using [Msa22,Thm. 3.1] to endow C with an extriangulated structure (F, t).…”

Section: Introductionmentioning

confidence: 99%

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Idempotent completions of $n$-exangulated categories

Klapproth¹,

Msapato²,

Shah³

2022

Preprint

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Suppose (C, E, s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C, E, s) to (resp. weakly) idempotent complete n-exangulated categories. We note that our methods of proof differ substantially from the extriangulated and (n + 2)-angulated cases. However, our constructions recover the known structures in the established cases up to nexangulated isomorphism of n-exangulated categories.Acknowledgements. The authors would like to thank Theo Bühler, Ruben Henrard and Adam-Christiaan van Roosmalen for useful email communications, and Andrew Brooke-Taylor and Peter Jørgensen for helpful discussions.

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Section: The Idempotent Completion Of An N-exangulated Categorymentioning

confidence: 86%

“…(iii) extriangulated (see [Msa22,Thm. 3.1]); and (iv) (n + 2)-angulated, where n 1 is an integer (see Lin [Lin21,Thm. 3.1]).…”

Section: Introductionmentioning

confidence: 99%

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Idempotent completions of $n$-exangulated categories

Klapproth¹,

Msapato²,

Shah³

2022

Preprint

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“…This attempt has been very successful, with many results that hold individually for exact and triangulated categories having been generalised to the extriangulated framework; see e.g. [32], [39], [47], [48], [50], amongst many others.…”

Section: Localisation In Triangulated Categoriesmentioning

confidence: 99%

The index with respect to a rigid subcategory of a triangulated category

Jørgensen¹,

Shah²

2022

Preprint

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Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero-Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index.Let C be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category (C, E, s). Suppose X is a contravariantly finite, rigid subcategory in C. We define the index ind X (C) of an object C ∈ C with respect to X as the K 0 -class [C] X in Grothendieck group K 0 (C, E X , s X ) of the relative extriangulated category (C, E X , s X ). By analogy to the classical case, we give an additivity formula with error term for ind X on triangles in C.In case X is contained in another suitable subcategory T of C, there is a surjectionThus, in order to describe K 0 (C, E X , s X ), it suffices to determine K 0 (C, E T , s T ) and Ker Q. We do this under certain assumptions.

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“…This new framework has proven significance: aside from permitting the unification and extension of many known results (see e.g. [20,21,49,52]), it has led to novel insights and explained previously mysterious connections (see e.g. [38,39,57]).…”

Section: Introductionmentioning

confidence: 99%

The category of extensions and a characterisation of $n$-exangulated functors

Bennett-Tennenhaus¹,

Haugland²,

Sandøy³

et al. 2022

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Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information. We study how functors between categories of extensions relate to those at the level of the original categories. When the additive categories in question are n-exangulated, this leads to a characterisation of n-exangulated functors.Our approach enables us to study n-exangulated categories from a 2-categorical perspective. We introduce n-exangulated natural transformations and characterise them using categories of extensions. Our characterisations allow us to establish a 2-functor between the 2-categories of small n-exangulated categories and small exact categories. A similar result with no smallness assumption is also proved.We employ our theory to produce various examples of n-exangulated functors and natural transformations. Although the motivation for this article stems from representation theory and the study of n-exangulated categories, our results are widely applicable: several require only an additive category equipped with a biadditive functor with no extra assumptions; others can be applied by endowing an additive category with its split n-exangulated structure.

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The Karoubi envelope and weak idempotent completion of an extriangulated category

Abstract: We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.

Cited by 9 publications

References 16 publications

Idempotent completions of $n$-exangulated categories

Idempotent completions of $n$-exangulated categories

The index with respect to a rigid subcategory of a triangulated category

The category of extensions and a characterisation of $n$-exangulated functors

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