The Cuboid Lemma and Mal’tsev Categories

详细信息    查看全文

• 作者：Marino Gran (1)
  Diana Rodelo (2) (3)
  
• 关键词：Goursat category ; 3 × 3 Lemma ; Pushouts ; Mal’tsev condition ; 18C05 ; 08C05 ; 18B10 ; 18E10
• 刊名：Applied Categorical Structures
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：22
• 期：5-6
• 页码：805-816
• 全文大小：275 KB
• 参考文献：1. Borceux, F., Bourn, D.: Mal’cev, protomodular, homological and semi-abelian categories. Kluwer (2004)
  2. Bourn, D.: The denormalized 3 × 3 lemma. J. Pur. Appl. Algebra 177, 113-29 (2003) CrossRef
  3. Carboni, A., Kelly, G.M., Pedicchio, M.C.: Some remarks on Maltsev and Goursat categories. Appl. Cat. Struct. 1, 385-21 (1993) 0872942" target="_blank" title="It opens in new window">CrossRef
  4. Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Mal’cev categories. J. Pur. Appl. Algebra 69, 271-84 (1991) CrossRef
  5. Everaert, T., Goedecke, J., Van der Linden, T.: Resolutions, higher extensions and the relative Mal’tsev axiom. J. Algebra 371, 132-55 (2012) CrossRef
  6. Everaert, T., Goedecke, J., Janelidze-Gray, T., Van der Linden, T.: Relative Mal’tsev categories. Theory Appl. Categories 28(29), 1002-021 (2013)
  7. Goedecke, J., Janelidze, T.: Relative Goursat categories. J. Pur. Appl. Algebra 216, 1726-733 (2012) CrossRef
  8. Gran, M., Janelidze, Z., Rodelo, D.: 3 × 3-Lemma for star-exact sequences. Homology Homotopy Appl. 14(2), 1-2 (2012) CrossRef
  9. Gran, M., Rodelo, D.: A new characterisation of Goursat categories. Appl. Categ. Struct. 20, 229-38 (2012) CrossRef
  10. Gran, M., Rossi, V.: Galois theory and double central extensions. Homology Homotopy Appl. 6(1), 283-98 (2004) CrossRef
  11. Janelidze, G.: What is a double central extension? Cah. Top. Géom. Diff. Catég XXXII(3), 191-01 (1991)
  12. Janelidze, T.: Foundation of relative non-abelian homological algebra, Ph.D. Thesis, University of Cape Town (2009)
  13. Janelidze, T.: Relative semi-abelian categories, Appl. Categ. Struct. 17, 373-86 (2009) 08-9155-2" target="_blank" title="It opens in new window">CrossRef
  14. Janelidze, Z.: The pointed subobject functor, 3 × 3 lemmas and subtractivity of spans. Th. Appl. Categ. 23, 221-42 (2010)
  15. Lack, S.: The 3-by-3 lemma for regular Goursat categories. Homology Homotopy Appl. 6(1), 1- (2004) CrossRef
  16. Smith, J.D.H.: Mal’cev Varieties, Lecture Notes in Math, vol. 554 (1976)
  
• 作者单位：Marino Gran (1)
  Diana Rodelo (2) (3)
  
  1. Insitut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium
  2. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-39, Faro, Portugal
  3. CMUC, Universidade de Coimbra, 3001-54, Coimbra, Portugal
  
• ISSN：1572-9095

文摘

We prove that a regular category ?is a Mal’tsev category if and only if a strong form of the denormalised 3 × 3 Lemma holds true in ? In this version of the 3 × 3 Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal’tsev categories (= 2-permutable) is similar to the one previously obtained for Goursat categories (= 3-permutable). We also analyse the “relative-version of the Cuboid Lemma and extend our results to that context.