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Relevant bibliographies by topics / Abelian categories

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Academic literature on the topic 'Abelian categories'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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Journal articles on the topic "Abelian categories"

1

Pop, Flaviu. "Coplexes in abelian categories." Studia Universitatis Babes-Bolyai Matematica 62, no. 1 (March 1, 2017): 3–13. http://dx.doi.org/10.24193/subbmath.2017.0001.

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Ebrahimi, Ramin, and Alireza Nasr-Isfahani. "Representation of n-abelian categories in abelian categories." Journal of Algebra 563 (December 2020): 352–75. http://dx.doi.org/10.1016/j.jalgebra.2020.07.010.

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ZHENG, QILIAN, and JIAQUN WEI. "QUOTIENT CATEGORIES OF n-ABELIAN CATEGORIES." Glasgow Mathematical Journal 62, no. 3 (September 30, 2019): 673–705. http://dx.doi.org/10.1017/s0017089519000417.

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AbstractThe notion of mutation pairs of subcategories in an n-abelian category is defined in this paper. Let ${\cal D} \subseteq {\cal Z}$ be subcategories of an n-abelian category ${\cal A}$. Then the quotient category ${\cal Z}/{\cal D}$ carries naturally an (n + 2) -angulated structure whenever $ ({\cal Z},{\cal Z}) $ forms a ${\cal D} \subseteq {\cal Z}$-mutation pair and ${\cal Z}$ is extension-closed. Moreover, we introduce strongly functorially finite subcategories of n-abelian categories and show that the corresponding quotient categories are one-sided (n + 2)-angulated categories. Finally, we study homological finiteness of subcategories in a mutation pair.

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Janelidze, George, László Márki, and Walter Tholen. "Semi-abelian categories." Journal of Pure and Applied Algebra 168, no. 2-3 (March 2002): 367–86. http://dx.doi.org/10.1016/s0022-4049(01)00103-7.

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Lyubashenko, V. "Ribbon Abelian Categories as Modular Categories." Journal of Knot Theory and Its Ramifications 05, no. 03 (June 1996): 311–403. http://dx.doi.org/10.1142/s0218216596000229.

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A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given a (eventually non-semisimple) k-linear abelian ribbon braided category [Formula: see text] with some finiteness conditions we construct a functor from a central extension of N with the set of labels [Formula: see text] ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surfto k-vector spaces.

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Amzil, Houda, Driss Bennis, J. R. García Rozas, Hanane Ouberka, and Luis Oyonarte. "Subprojectivity in Abelian Categories." Applied Categorical Structures 29, no. 5 (March 11, 2021): 889–913. http://dx.doi.org/10.1007/s10485-021-09638-w.

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Gillespie, James. "Hereditary abelian model categories." Bulletin of the London Mathematical Society 48, no. 6 (September 13, 2016): 895–922. http://dx.doi.org/10.1112/blms/bdw051.

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Zhou, Panyue, and Bin Zhu. "n-Abelian quotient categories." Journal of Algebra 527 (June 2019): 264–79. http://dx.doi.org/10.1016/j.jalgebra.2019.03.007.

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Kálnai, Peter, and Jan Žemlička. "Compactness in abelian categories." Journal of Algebra 534 (September 2019): 273–88. http://dx.doi.org/10.1016/j.jalgebra.2019.05.037.

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Chen, Xiao-Wu, and Henning Krause. "Expansions of abelian categories." Journal of Pure and Applied Algebra 215, no. 12 (December 2011): 2873–83. http://dx.doi.org/10.1016/j.jpaa.2011.04.008.

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Dissertations / Theses on the topic "Abelian categories"

1

Pettersson, Samuel. "Additive, abelian, and exact categories." Thesis, Uppsala universitet, Algebra och geometri, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-312384.

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Steine, Asgeir Bertelsen. "STABILITY STRUCTURES FOR ABELIAN AND TRIANGULATED CATEGORIES." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2007. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9603.

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This thesis is intended to present some developments in the theory of algebraic stability. The main topics are stability for triangulated categories and the distinguished slopes of Hille and de la Pena for quiver representations.

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McBride, Aaron. "Grothendieck Group Decategorifications and Derived Abelian Categories." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/33000.

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The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorication from the category of essentially small triangulated categories (whose morphisms are triangulated functors) to the category of abelian groups. In the case of an essentially small abelian category, its Grothendieck group and the triangulated Grothendieck group of its bounded derived category are isomorphic as groups via a natural map. Because of this, homological algebra and derived functors become useful in surprising ways. This thesis is an expository work that provides an overview of the theory of Grothendieck groups with respect to these decategorications.

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Abdulwahid, Adnan Hashim. "Cofree objects in the categories of comonoids in certain abelian monoidal categories." Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/2032.

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We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We nd concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in [4] on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.

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Shepherd, James A. "Coherent sheaves and deformation theory in abelian categories." Thesis, University of Bristol, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.412375.

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Ahlsén, Daniel. "Classifying Categories : The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-352383.

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Bridge, Philip Owen. "Essentially algebraic theories and localizations in toposes and abelian categories." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/essentially-algebraic-theories-and-localizations-in-toposes-and-abelian-categories(2db96543-4a42-49fe-8741-ffa1ff249b12).html.

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The main theme of this thesis is the parallel between results in topos theory and the theory of additive functor categories. In chapter 2, we provide a general overview of the topics used in the rest of the thesis. Locally finitely presentable categories are introduced, and their expression as essentially algebraic categories is explained. The theory of localization for toposes and abelian categories is introduced, and it is shown how these localizations correspond to theories in appropriate logics. In chapter 3, we look at conditions under which the category of modules for a ring object R in a topos E is locally finitely presented, or locally coherent. We show that if E is locally finitely presented, then the category of modules is also; however we show that far stronger conditions are required for the category of modules to be locally coherent. In chapter 4, we show that the Krull-Gabriel dimension of a locally coherent abelian category C is equal to the socle length of the lattice of regular localizations of C. This is used to make an analogous definition of Krull-Gabriel dimension for regular toposes, and the value of this dimension is calculated for the classifying topos of the theory of G-sets, where G is a cyclic group admitting no elements of square order. In chapter 5, we introduce a notion of strong flatness for algebraic categories (in the sense studied by Adamek, Rosickey and Vitale). We show that for a monoid M of finite geometric type, or more generally a small category C with the corresponding condition, the category of M-acts, or more generally the category of set-valued functors on C, has strongly flat covers.

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Goedecke, Julia. "Three viewpoints on semi-abelian homology." Thesis, University of Cambridge, 2009. https://www.repository.cam.ac.uk/handle/1810/224397.

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The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.

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Krämer, Thomas [Verfasser], and Rainer [Akademischer Betreuer] Weissauer. "Tannakian Categories of Perverse Sheaves on Abelian Varieties / Thomas Krämer ; Betreuer: Rainer Weissauer." Heidelberg : Universitätsbibliothek Heidelberg, 2013. http://d-nb.info/1177249111/34.

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Michael, Ifeanyi Friday. "On a unified categorical setting for homological diagram lemmas." Thesis, Stellenbosch : Stellenbosch University, 2011. http://hdl.handle.net/10019.1/18085.

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Thesis (MSc)--Stellenbosch University, 2011.
ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis.
AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.

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Books on the topic "Abelian categories"

1

Borceux, Francis. Mal'cev, protomodular, homological and semi-abelian categories. Dordrecht: Kluwer Academic, 2004.

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2

1942-, Reiten Idun, and Smalø Sverre O, eds. Tilting in Abelian categories and quasitilted algebras. Providence, R.I: American Mathematical Society, 1996.

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Borceux, Francis, and Dominique Bourn. Mal’cev, Protomodular, Homological and Semi-Abelian Categories. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3.

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4

Borceux, Francis. Handbook of categorical algebra 2: Categories and structures. Cambridge [England]: Cambridge University Press, 1994.

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5

Definable additive categories: Purity and model theory. Providence, R.I: American Mathematical Society, 2011.

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6

Verdier, Jean Louis. Des catégories dérivées des catégories abéliennes. Paris: Société mathématique de France, 1996.

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7

Bernstein, Joseph. Equivariant sheaves and functors. Berlin: Springer-Verlag, 1994.

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8

1943-, Schapira Pierre, ed. Ind-sheaves. Paris: Société mathématique de France, 2001.

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9

A non-Hausdorff completion: The Abelian category of C-complete left modules over a topological ring. New Jersey: World Scientific, 2015.

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10

Pantev, Tony. Stacks and catetories in geometry, topology, and algebra: CATS4 Conference Higher Categorical Structures and Their Interactions with Algebraic Geometry, Algebraic Topology and Algebra, July 2-7, 2012, CIRM, Luminy, France. Providence, Rhode Island: American Mathematical Society, 2015.

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Book chapters on the topic "Abelian categories"

1

Holme, Audun. "Abelian Categories." In A Royal Road to Algebraic Geometry, 165–83. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-19225-8_8.

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Borceux, Francis, and Dominique Bourn. "Semi-abelian categories." In Mal’cev, Protomodular, Homological and Semi-Abelian Categories, 319–43. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3_6.

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Penner, Robert. "Abelian and Additive Categories." In Lecture Notes in Mathematics, 51–55. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43996-5_10.

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Hanratty, Chantelle. "Abelian and Triangulated Categories." In Springer Proceedings in Mathematics & Statistics, 3–16. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91626-2_1.

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Zimmermann, Alexander. "Abelian and Triangulated Categories." In Algebra and Applications, 259–385. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-07968-4_3.

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Borceux, Francis, and Dominique Bourn. "Mal’cev categories." In Mal’cev, Protomodular, Homological and Semi-Abelian Categories, 125–228. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3_3.

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Borceux, Francis, and Dominique Bourn. "Protomodular categories." In Mal’cev, Protomodular, Homological and Semi-Abelian Categories, 229–71. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3_4.

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Borceux, Francis, and Dominique Bourn. "Homological categories." In Mal’cev, Protomodular, Homological and Semi-Abelian Categories, 273–317. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3_5.

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Penner, Robert. "Diagram Chasing in Abelian Categories." In Lecture Notes in Mathematics, 57–60. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43996-5_11.

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Borceux, Francis, and Dominique Bourn. "Strongly protomodular categories." In Mal’cev, Protomodular, Homological and Semi-Abelian Categories, 345–70. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3_7.

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Conference papers on the topic "Abelian categories"

1

Mucuk, Osman, and Serap Demir. "Internal categories in the category of semi abelian algebras." In FOURTH INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2020). AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0042238.

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