Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories

In this article, we study the heart of a cotorsion pairs on an exact category and a triangulated category in a unified meathod, by means of the notion of an extriangulated category. We prove that the heart is abelian, and construct a cohomological functor to the heart. If the extriangulated category has enough projectives, this functor gives an equivalence between the heart and the category of coherent functors over the coheart modulo projectives. We also show how an n-cluster tilting subcategory of an extriangulated category gives rise to a family of cotorsion pairs with equivalent hearts.

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“…In the rest of this article, we fix an extriangulated category (B, E, s). The following have been shown in [NP,Propositions 3.3,3.11,3.15].…”

Section: Introductionmentioning

confidence: 94%

“…Remark 1.16. By [NP,Corollary 3.8], for any E-triangle A Definition 1.17. Let D ⊆ B be a full additive subcategory, closed under isomorphisms.…”

Section: Introductionmentioning

confidence: 99%

Hearts of twin cotorsion pairs on extriangulated categories

2019

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“…Hovey TCP, as a generalization of recollement. In view of [Ho1], [Ho2], the following has been defined in [NP,Definition 5.1].…”

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confidence: 99%

A simultaneous generalization of mutation and recollement on a triangulated category

Nakaoka

2015

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In this article, we introduce the notion of concentric twin cotorsion pair on a triangulated category. This notion contains the notions of tstructure, cluster tilting subcategory, co-t-structure and functorally finite rigid subcategory as examples. Moreover, a recollement of triangulated categories can be regarded as a special case of concentric twin cotorsion pair.To any concentric twin cotorsion pair, we associate a pretriangulated subquotient category. This enables us to give a simultaneous generalization of the Iyama-Yoshino reduction and the recollement of cotorsion pairs. This allows us to give a generalized mutation on cotorsion pairs defined by the concentric twin cotorsion pair.

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“…Definition 2.3. [NP,Definition 2.12] We call the pair (E, s) an external triangulation of C if it satisfies the following conditions:…”

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confidence: 99%

Triangulated quotient categories revisited

2018

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Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this article. Let A be an extension closed subcategory of an extriangulated category C . Then the quotient category M := A/X carries naturally a triangulated structure whenever (A, A) forms an X -mutation pair. This result unifies many previous constructions of triangulated quotient categories, and using it gives a classification of thick triangulated subcategories of pretriangulated category C /X , where X is functorially finite in C . When C has Auslander-Reiten translation τ , we prove that for a functorially finite subcategory X of C containing projectives and injectives, C /X is a triangulated category if and only if (C , C ) is X −mutation, and if and only if τ X = X . This generalizes a result by Jørgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory X of the extriangulated category C , C admits a new extriangulated structure such that C is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present examples that extriangulated categories are neither exact categories nor triangulated categories.

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Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories

Cited by 12 publications

References 10 publications

Hearts of twin cotorsion pairs on extriangulated categories

Hearts of twin cotorsion pairs on extriangulated categories

A simultaneous generalization of mutation and recollement on a triangulated category

Triangulated quotient categories revisited

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