A Generalization of Gabriel’s Galois Covering Functors II: 2-Categorical Cohen-Montgomery Duality

Asashiba, Hideto

doi:10.1007/s10485-015-9416-9

Cited by 12 publications

(17 citation statements)

References 13 publications

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“…Then we consider moving between the two notions of grading using the well-known operations of orbit categories and smash products. We use Asashiba's theorem that these are 2-functors and they give an equivalence of 2-categories [Asa17].…”

Section: Summary Of Contentsmentioning

confidence: 99%

“…We largely follow Asashiba [Asa17], though our conventions are slightly different: see Remark 4.24 below. Orbit categories rose in prominence after their use in categorifying cluster algebras by Buan, Marsh, Reineke, Reiten, and Todorov [BMRRT06] and related work of Keller [Kel05].…”

Section: Graded Categoriesmentioning

confidence: 99%

“…Remark 4.24. Our categories are defined oppositely to [Asa17], where the natural transformations given as part of an equivariant structure are in the opposite direction. The orbit category and smash product category are defined with opposite signs, which correspond to our opposite conventions: see [Asa11, Proposition 2.11].…”

Section: Asashiba's Biequivalencementioning

confidence: 99%

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Serre functors and graded categories

Grant¹

2020

Preprint

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We study Serre structures on two types of graded -linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This allows us to show that Iyama's higher Auslander algebras of type A are fractionally Calabi-Yau.

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Section: Summary Of Contentsmentioning

confidence: 99%

Section: Graded Categoriesmentioning

confidence: 99%

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Serre functors and graded categories

Grant¹

2020

Preprint

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“…(4) Give a condition on a 1-morphism between colax functors to be an equivalence. (5) Give a natural definition of a derived equivalence between colax functors by the equivalence (defined in (3)) of their "derived categories" defined in (2). (6) Characterize the existence of derived equivalences of colax functors by tilting subcategories, which turns out to be a generalization of Rickard's Morita theorem for colax functors.…”

Section: Introductionmentioning

confidence: 99%

“…In the last section we give a proof of Theorem 6.5. pitality. Finally, I would like to thank D. Tamaki for useful discussions with him on Grothendieck constructions and for his expositions on 2-categorical notions through his preprints [15,16] that aimed at a generalization of [5]. In addition I would also like to thank the referee for his/her careful reading, suggestions and questions, by which the paper became easier to read and I could notice that I forgot to consider the naturality property (0) of 1-morphisms in Definition 6.1 and I could add the verification of this property in the proof of Lemma 9.3; also I changed the terminology "oplax" to "colax".…”

Section: Introductionmentioning

confidence: 99%

Gluing derived equivalences together

Asashiba

2013

Advances in Mathematics

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The Grothendieck construction of a diagram X of categories can be seen as a process to construct a single category Gr(X) by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category I to the 2-category k-Cat of small k-categories for a fixed commutative ring k. In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors X, X ′ : I → k-Cat are derived equivalent if there is a derived equivalence from X(i) to X ′ (i) for all objects i in I satisfying some "Iequivariance" conditions. In this paper we glue the derived equivalences between X(i) and X ′ (i) together to obtain a derived equivalence between Grothendieck constructions Gr(X) and Gr(X ′ ), which shows that if colax functors are derived equivalent, then so are their Grothendieck constructions. This generalizes and well formulates the fact that if two k-categories with a G-action for a group G are "G-equivariantly" derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two k-algebras A and A ′ are derived equivalent, then so are the path categories AQ and A ′ Q for any quiver Q; so are the incidence categories AS and A ′ S for any poset S; and so are the monoid algebras AG and A ′ G for any monoid G. Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence.

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On Frobenius (Completed) Orbit Categories

Chávez

2017

Algebr Represent Theor

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Abstract. Let E be a Frobenius category, P its subcategory of projective objects and F : E → E an exact automorphism. We prove that there is a fully faithful functor from the orbit category E/F into gpr(P/F ), the category of finitely-generated Gorensteinprojective modules over P/F . We give sufficient conditions to ensure that the essential image of E/F is an extension-closed subcategory of gpr(P/F ). If E is in addition KrullSchmidt, we give sufficient conditions to ensure that the completed orbit category E /F is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.

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A Generalization of Gabriel’s Galois Covering Functors II: 2-Categorical Cohen-Montgomery Duality

Cited by 12 publications

References 13 publications

Serre functors and graded categories

Serre functors and graded categories

Gluing derived equivalences together

On Frobenius (Completed) Orbit Categories

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