Triangulated quotient categories revisited

Zhou, Panyue; Zhu, Bin

doi:10.48550/arxiv.1608.00297

Cited by 2 publications

(2 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To overcome the difficulty, Nakaoka and Palu [15] introduced the notion of externally triangulated categories (extriangulated categories for short) by a careful looking what is necessary in the definition of cotorsion pairs in exact and triangulated cases. Under this notion, exact categories and extension-closed subcategories of triangulated categories both are externally triangulated, and hence, in some levels, it becomes easy to give uniform statements and proofs for the exact and triangulated settings [15,20].…”

Section: Introductionmentioning

confidence: 99%

Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

Zhao¹,

Huang²

2019

Taiwanese J. Math.

View full text Add to dashboard Cite

In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if (C , E, s) is an extriangulated category with enough injective objects and projective objects, then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of C ; (2) special preenveloping ideals of C ; (3) additive subfunctors of E having enough special injective morphisms; and (4) additive subfunctors of E having enough special projective morphisms. Moreover, we show that if (C , E, s) is an extriangulated category with enough injective objects and projective morphisms, then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of C ; (2) all additive subfunctors of E having enough special injective objects.

show abstract

Section: Introductionmentioning

confidence: 99%

Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

Zhao¹,

Huang²

2019

Taiwanese J. Math.

View full text Add to dashboard Cite

show abstract

“…In particular, the identity map is a bijection from the set of cluster-tilting objects of E + Q to the set of cluster-tilting objects of E Q , commuting with mutation. A more detailed discussion of mutation of cluster-tilting objects in extriangulated categories can be found in [13] (see also [41]). Now let T be a cluster-tilting object of…”

Section: An Extriangulated Categorificationmentioning

confidence: 99%

A categorification of acyclic principal coefficient cluster algebras

Pressland¹

2017

Preprint

View full text Add to dashboard Cite

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables, requiring as input a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category. In this paper, we construct such candidates in the case of cluster algebras with 'polarised' principal coefficients, and obtain Frobenius categorifications in the acyclic case. Since cluster algebras with principal coefficients are obtained from those with polarised principal coefficients by setting half of the frozen variables to 1, our categories also indirectly model principal coefficient cluster algebras, for which no Frobenius categorification exists in general. Moreover, partial stabilisation provides an extriangulated category, in the sense of Nakaoka and Palu, that may more directly model the principal coefficient cluster algebra. Many of the intermediate results remain valid without the acyclicity assumption, as we will indicate. Along the way, we establish a Frobenius version of Keller's result that the Ginzburg dg-algebra of a quiver with potential is bimodule 3-Calabi-Yau.

show abstract

Triangulated quotient categories revisited

Cited by 2 publications

References 13 publications

Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

A categorification of acyclic principal coefficient cluster algebras

Contact Info

Product

Resources

About