ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Kopylov Y. A. On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 4, pp. 434-443. DOI: 10.18500/1816-9791-2020-20-4-434-443

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On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories

Kopylov Yaroslav A., Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences (Sobolev Institute of Mathematics)

As is well known, many important additive categories in functional analysis and algebra are not abelian. Many classical diagram assertions valid in abelian categories fail in more general additive categories without additional assumptions concerning the properties of the morphisms of the diagrams under consideration. This in particular applies to the so-called Snake Lemma, or the KerCoker-sequence. We obtain a theorem about a diagram generalizing the classical situation of the Snake Lemma in the context of categories semi-abelian in the sense of Palamodov. It is also known that, already in P-semi-abelian categories, not all kernels (respectively, cokernels) are semi-stable, that is, stable under pushouts (respectively, pullbacks). We prove a proposition showing how non-semi-stable kernels and cokernels can arise in general preabelian categories.

  1. Kopylov Ya. A., Kuz′minov V. I. On the Ker-Coker-sequence in a semiabelian category. Siberian Math. J., 2000, vol. 41, no. 3, pp. 509–517. DOI:
  2. Kopylov Ya. A., Kuz′minov V. I. The Ker-Coker-sequence and its generalization in some classes of additive categories. Siberian Math. J., 2009, vol. 50, no. 1, pp. 86–95. DOI:
  3. Grandis M. On the categorical foundations of homological and homotopical algebra. Cah. Topol. G´eom. Diff´er. Cat´eg., 1992, vol. 33, no. 2, pp. 135–175.
  4. Bucur I., Deleanu A. Introduction to the Theory of Categories and Functors. London, New York, Sydney, Interscience Publ., John Wiley & Sons, Ltd., 1968. 224 p.
  5. Raıkov D. A. Semiabelian categories. Soviet Math. Dokl., 1969, vol. 10, pp. 1242–1245.
  6. Palamodov V. P. Homological methods in the theory of locally convex spaces. Russ. Math. Surv., 1971, vol. 26, iss. 1, pp. 1–64. DOI:
  7. Nomura Y. Induced morphisms for Lambek invariants of commutative squares. Manuscr. Math., 1971, vol. 4, iss. 3, pp. 263–275. DOI:
  8. Eckmann B., Hilton P. J. Exact couples in an abelian category. J. Algebra, 1966, vol. 3, pp. 38–87. DOI:
  9. Kuz′minov V. I., Cherevikin A. Yu. Semiabelian categories. Siberian Math. J., 1972, vol. 13, no. 6, pp. 895–902. DOI:
  10. Yakovlev A. V. Homological algebra in pre-Abelian categories. J. Math. Sci., 1982, vol. 19, iss. 1, pp. 1060–1067. DOI:
  11. Rump W. Almost abelian categories. Cah. Topol. G´eom. Diff´er. Cat´eg., 2001, vol. 42, no. 3, pp. 163–225.
  12. Kopylov Ya. A., Wegner S.-A. On the notion of a semi-abelian category in the sense of Palamodov. Appl. Categor. Struct., 2012, vol. 20, pp. 531–541. DOI:
  13. Schneiders J.-P. Quasi-abelian categories and sheaves. M´emoires de la Soci´et´e Math´ematique de France, Ser. 2, 1999, no. 76, 144 p. DOI:
  14. Bonet J., Dierolf S. The pullback for bornological and ultrabornological spaces. Note Mat., 2006, vol. 25, no. 1, pp. 63–67. DOI:
  15. Rump W. A counterexample to Ra˘ıkov’s conjecture. Bull. Lond. Math. Soc., 2008, vol. 40, iss. 6, pp. 985–994. DOI:
  16. Rump W. Analysis of a problem of Raikov with applications to barreled and bornological spaces. J. of Pure Appl. Algebra, 2011, vol. 215, iss. 1, pp. 44–52. DOI:
  17. Wengenroth J. The Raikov conjecture fails for simple analytical reasons. J. Pure Appl. Algebra, 2012, vol. 216, iss. 7, pp. 1700–1703. DOI:
  18. Kelly G. M. Monomorphisms, epimorphisms, and pull-backs. J. Austral. Math. Soc., 1969, vol. 9, pp. 124–142. DOI:
  19. Gelfand I. M., Manin Yu. I. Methods of Homological Algebra. Springer Monographs in Mathematics. Berlin, Springer-Verlag, 2003. 372 p.
  20. Kopylov Ya. A., Kuz′minov V. I. Exactness of the cohomology sequence corresponding to a short exact sequence of complexes in a semiabelian category. Siberian Adv. Math., 2003, vol. 13, no. 3, pp. 72–80.
  21. Kopylov Ya. A. Homology in P-semi-abelian categories. Sci. Ser. A Math. Sci. (N.S.), 2009, vol. 17, pp. 105–114.
  22. Kopylov Ya. A. On the homology sequence in a P-semi-abelian category. Sib. Elektron. Mat. Izv., 2012, vol. 9, pp. 190–200.
  23. Makarov B. M. Some pathological properties of inductive limits of B-spaces. Uspekhi Mat. Nauk, 1963, vol. 18, iss. 3 (111), pp. 171–178 (in Russian).