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Hereditary uniserial categories with Serre duality
TL;DR: In this article, the authors classify the hereditary uniserial categories with Serre duality into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant).
Abstract: An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.
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TL;DR: In this paper, Buan et al. introduce continuous Frobenius categories, which are topological categories which are constructed using representations of the circle over a discrete valuation ring, and they show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle.
Abstract: We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle. By putting restrictions on these points we obtain various Frobenius subcategories. The main purpose of constructing these Frobenius categories is to give a precise and elementary description of the triangulated structure of their stable categories. We show in Igusa and Todorov (arXiv:1209.1879, 2012) for which parameters these stable categories have cluster structure in the sense of Buan et al. (Compos. Math. 145:1035–1079, 2009) and we call these continuous cluster categories.
25 citations
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TL;DR: In this paper, it was shown that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The continuous cluster categories of arXiv:1209.1879 are examples of this construction.
Abstract: Cyclic poset are generalizations of cyclically ordered sets. In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The continuous cluster categories of arXiv:1209.1879 are examples of this construction. If we twist the construction using an admissible automorphism of the cyclic poset, we generate other examples such as the m-cluster category of type A-infinity (m>2).
20 citations
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TL;DR: In this paper, the authors constructed topological triangulated categories C_c as stable categories of certain topological Frobenius categories F_c and showed that these categories have a cluster structure for certain values of c including c=pi.
Abstract: In arXiv:1209.0038 we constructed topological triangulated categories C_c as stable categories of certain topological Frobenius categories F_c. In this paper we show that these categories have a cluster structure for certain values of c including c=pi. The continuous cluster categories are those C_c which have cluster structure. We study the basic structure of these cluster categories and we show that C_c is isomorphic to an orbit category D_r/F_s of the continuous derived category D_r if c=r pi/s. In C_pi, a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.
19 citations
Cites background from "Hereditary uniserial categories wit..."
...The abelian category A. The category AR will be defined to be the category of representations of the “quiver” R. This is similar to the representations of infinite linearly ordered posets considered in [15], except that R is not locally finite, so we do not have Serre duality. Definition 1.1.1. A representation of R over a field k is defined to be a functor Vwhich assigns a k -vector space Vx to every x∈R a...
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...call this the basic morphism from P[x] to P[y]. All morphisms of the form e2π are multiplication by the uniformizer t∈Rby definition. See [8] for details. (See also the construction of the big loop in [15].) Let Fbe the category of all pairs (V,d) where Vis an object of PS1 and dis an endomorphism of V so that d2 = ·t (multiplication by t). Morphisms (V,d) →(W,d) are defined to be morphisms f: V →Win PS...
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TL;DR: In this article, it was shown that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver.
Abstract: We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality.
7 citations
Cites background from "Hereditary uniserial categories wit..."
...Let Q be the thread quiver given by x y The category rep Q has been discussed in [26], where it is shown that repQ contains a so-called big tube (defined in [26])....
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TL;DR: In this article, the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank with Serre duality, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.
Abstract: Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.
5 citations
Cites background from "Hereditary uniserial categories wit..."
...For all of the three classes in Theorem 1, infinite versions have been constructed in [8, 39, 45]....
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...We recall the following result from [45] (based on [27])....
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References
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Book•
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
9,254 citations
"Hereditary uniserial categories wit..." refers methods in this paper
...Following [19, 29] we will write ◦ for vertical composition of 2-cells and • for horizontal composition....
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Book•
01 Jan 1982
TL;DR: Lack, Ross Street and Wood as discussed by the authors present a mathematical subject classification of 18-02, 18-D10, 18D20, and 18D21 for mathematics subject classification.
Abstract: Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.
1,492 citations
"Hereditary uniserial categories wit..." refers methods in this paper
...We will repeat some definitions and results from [29] (see also [5, 6, 17])....
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TL;DR: The Bulletin de la S. M. F. as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.html).
Abstract: © Bulletin de la S. M. F., 1962, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
1,474 citations
"Hereditary uniserial categories wit..." refers background in this paper
...It is known [11] that every locally finite Grothendieck category is dual to a category of pseudocompact modules over a pseudocompact ring....
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...A pseudocompact ring ([11]) which is also a k-algebra is not necessarily a pseudocompact k-algebra....
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...In [11] it has been shown that every locally finite Grothendieck category (thus not necessarily of finite type) is dual to the category PC(R) for a pseudocompact ring....
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...It has been shown in [11] that IndA is a Grothendieck category....
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...We refer to [11] for more information on pseudocompact rings, and to [7] for information on coalgebras and comodules....
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26 Aug 1994
TL;DR: The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen.
Abstract: The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.
1,321 citations
Additional excerpts
...We will repeat some definitions and results from [29] (see also [5, 6, 17])....
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31 Jan 2022
TL;DR: In this paper , the authors give a selfcontained account of basic category theory as described above, assuming as prior knowledge only the most elementary categorical concepts, and treating the ordinary and enriched cases together from Chapter 3 on.
Abstract: Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it. As the applications of the theory continue to expand - some recent examples are given below - the lack of such an account is the more acutely felt. The present book is designed to supply the want in part, by giving a fairly complete treatment of the limited area to which the title refers. The basic concepts of category theory certainly include the notion of functor-category, of limit and colimit, of Kan extension, and of density; with their applications to completions, perhaps including those relative completions given by categories of algebras for limit-defined theories. If we read 'V-category' for 'category' here, this is essentially the list of our chapter-headings below, after the first chapter introducing V-categories. In fact our scope is wider than this might suggest; for what we give is also a selfcontained account of basic category theory as described above, assuming as prior knowledge only the most elementary categorical concepts, and treating the ordinary and enriched cases together from Chapter 3 on.
1,078 citations