nLab Malcev, protomodular, homological and semi-abelian categories

This entry is about the book:

• Francis Borceux, Dominique Bourn,

  Mal’cev, protomodular, homological and semi-abelian categories

  Mathematics and Its Applications 566,

  Kluwer 2004

  doi:10.1007/978-1-4020-1962-3

on nonabelian homological algebra, such as its diagram chasing lemmas, in Mal'cev categories, protomodular categories, semi-abelian categories and homological categories.

From the introduction:

The most striking successes of category theory, as far as clarification of mathematical situations is concerned, are probably the theory of abelian categories and the theory of toposes. This is not too amazing since both theories are closely related to the development of sheaf theory, a context in which it is desirable to get rid of the usual notion of element.

But up to recently, category theory did not provide any comparable insight in General Algebra, a domain in which element-based mathematics remains the slogan. In particular, category theory could not provide a structural tool able to grasp, even in the most representative category of classical algebra - namely, the category $\mathrm{Gp}$ of groups - the deep essence of the notion of normal subobject: namely, an equivalence class for a congruence and not just the kernel of a morphism.

And category theory could not grasp either the conceptual foundations of the homological lemmas: the Nine Lemma, the Snake Lemma, which remain valid and strongly meaningful in the category $\mathrm{Gp}$ of groups, even if this category does not belong to the abelian setting in which these lemmas are generally proved in a significant categorical way.

Of course, there have been since a long time attempts to provide an axiomatic context in which to get the isomorphism theorems, the decomposition theorems or the previous homological lemmas for the varieties of Universal Algebra: Baer (1947, [6]), Goldie (1952, [48]), Atiyah (1956, [5]), Higgins (1956, [54]), Kurosh (1959, [73]), Hilton-Ledermann (1960, [55]), Eckmann-Hilton (1962, [40]), Tsalenko (1967, [92]), but also Hofmannn (1960, [56]), Fröhlich (1961, [46]), Huq (1968, [57]), Gerstenhaber (1970, [47]), Burgin (1970, [34]), Orzech (1972, [83]).

These first attempts, despite their interest, consist generally in a long list of axioms whose independence is certainly not clear. But more importantly, these axioms look desperately heavy and complicated in comparison with the elegance of the characterization of abelian models. We refer the reader to the introduction of the paper by Janelidze-Márki-Tholen (2002, [60]) for a reliable historical approach to this topic. … Establishing an organic and synthetic connection between all these attempts the ambition of this book. To achieve this, an additional ingredient was necessary, of purely categorical nature: the fibration of points. This fibration allows representing every category as a fibration whose fibres are pointed categories, i.e. categories with a zero object (see Bourn, 1996, [17]). This book will give evidence that the fibration of points emphasizes the importance of split epimorphisms in the context of algebraic theories, but also that this fibration of points has a very strong classification power: see on page 466 the table summarizing these classification properties. …

Contents

(with links to the related $n$lab entries)

• Preface

1. Metatheorems

2. Mal'cev categories

3. Protomodular categories

4. Homological categories

5. Semi-abelian categories

6. Strongly protomodular categories

7. Essentially affine categories

• Appendix

  1. Algebraic theories

  2. Internal relations

  3. Internal groupoids

  4. Variations on epimorphisms

  5. Regular and exact categories

  6. Monads

  7. Fibrations

• Classification table of fibrations of points

• Bibliography

• Index of symbols

• Index of definitions

Bibliography

1. P. Agliano, A. Ursini, Ideals and other generalizations of congruence classes, Journal of Australian Math. Soc., Ser. A 53 (1992) 103-115

2. P. Agliano, A. Ursini, On subtractive varieties II: general properties, Algebra Universalis 36 (1996) 222-259

3. P. Agliano, A. Ursini, On subtractive varieties III: from ideals of congruences, Algebra Universalis 37 (1997) 296-333

4. P. Agliano, A. Ursini, On subtractive varieties IV: definability of principal ideals, Algebra Universalis 38 (1997) 355–389

5. M. Atiyah, On the Krull-Schmidt Theorem with applications to sheaves, Bull. Soc. Math. France 84 (1956) 307-317

6. R. Baer, Direct decomposition, Trans. Amer. Math. Soc. 62 (1947) 62-98

7. M. Barr, Exact categories, Springer Lect. Notes in Math. 236 (1971) 1-120

8. M. Barr, J. Beck, Hornology and standard constructions, Springer Lec. Notes in Math. 80 (1969) 245-335

9. J. Bénabou, Introduction to bicategories, Springer Lect. Notes in Math. 47 (1967) 1-77

10. J. Bénabou, Fibered categories and the foundations of naive category theory, J. of Symbolic Logic 50 (1985) 10

11. F. Borceux, Handbook of Categorical Algebra, Cambridge Univ. Press, vol. 1-3 (1994)

12. F. Borceux, A survey of semi-abelian categories, in: Galois theory, Hopf algebras, and semiabelian categories, 27–60, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004, MR2005b:18015

13. F. Borceux, M. M. Clementino, Topological semi-abelian algebras, Adv. Math. 190 (2005), no. 2, 425–453, doi, MR2005b:18015

14. F. Borceux, M. Grandis, Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra 208 (2007), no. 2, 665–689, doi, MR2007k:18021

15. D. Bourn, The shift functor and the comprehensive factorization for internal groupoids, Cahiers Top. Géométrie Différentielle Catégoriques 28 (1987) 197-226

16. D. Bourn, Normalization equivalence, kernel equivalence and alefine categories, Springer Lect. Notes in Math. 1448 Como (1991) 43-62

17. D. Bourn, Mal’cev categories and fibrations of pointed objects, Appl. Categorical Structures 4 (1996) 302-327

18. D. Bourn, Baer sums and fibered aspects of Mal’cev operations, Cahiers Top. Géom. Diff. Catégoriques 40 (1999) 297-316

19. D. Bourn, Normal subobjects and abelian objects in protomodular categories. J. Algebra 228 (2000) 143-164

20. D. Bourn, Normal functors and strong protomodularity, Theory Appl. Cat- egories 7 (2000) 206-218

21. D. Bourn, 3 x 3 lemma and protomodularity, J. Algebra 236 (2001) 778-795

22. D. Bourn, A categorical genealogy for the congruence distributive property, Theory Appl. Categories 8 (2001) 391-407

23. D. Bourn, Intrinsic centrality and associated classifying properties, J. of Algebra 256 (2002) 126-145

24. D. Bourn, Aspherical abelian groupoids and their directions, J. Pure Appl. Alg. 168 (2002) 133 146

25. D Bourn, The denormalized 3 x 3 lemma, J. Pure Appl. Algebra 177 (2003) 113-129

26. D Bourn, Commutator theory in regular Mal’cev categories, Publications of the Fields Institute (to appear)

27. D Bourn, Protomodular aspect of the dual of a topos, Advances in Mathematics, Adv. Math. 187 (2004), no. 1, 240–255, doi, MR2006a:18003

28. D Bourn, Commutator theory in strongly semi-abelian categories, Preprint, Univ. du Littoral (2003), submitted for publication

29. D Bourn, M. Gran, Centrality and normality in protomodular categories, Theory Appl. Categories 9 (2002) 151-165

30. D Bourn, M. Gran, Central extensions in semi-abelian categories, J. Pure Appl. Alg. 175 (2002) 31-44

31. D. Bourn, M. Gran, Centrality and connectors in Maltsev categories, Algebra Universalis 48 (2002) 309-331

32. D. Bourn, G. Janelidze, Protomodularity, descent and semi-direct product, Theory Appl. Categories 4 (1998) 37-46

33. D. Bourn, G. Janelidze, Characterization of protomodular varieties of universal algebra, Theory Appl. Categories 11 (2003) 143-147

34. M.S. Burgin, Categories with involution and correspondences in $\gamma$-categories, Trans. Moscow Math. Soc. 22 (1970) 181-257

35. A. Carboni, Categories of affine spaces, J. Pure Appl. Alg. 61 (1989) 243-25O

36. A. Carboni, G. M. Kelly, M. C. Pedicchio, Some remarks on Mal’tsev and Goursat categories, Appl. Categorical Structures 1 (1993) 385-421

37. A. Carboni, J. Lambek, M. C. Pedicchio, Diagram chasing in Mal’cev categories, J. Pure Appl. Alg. 69 (1990) 271-284

38. A. Carboni, M. C. Pedicchio, N. Pirovano, Internal graphs and internal groupolds in Mal’cev categories, CMS Conference proceedings, Category Theory 1991 13 (1992) 97-109

39. Y. Diers, Categories of commutative algebras, Oxford University Press (1992)

40. B. Eckmann, P.J. Hilton, Group-like structures in general categories Math. Ann. 145 (1962) 227-255

41. T. Everaert, T. Van der Linden, Baer invariants in semi-abelian categories H: homologs, preprint (2003)

42. T.H. Fay, On categorical conditions for congruences to commute, Algebra Univ. 8 (1978) 173-179

43. R. Freese, R. McKenzie, Commutator theory for congruence modular varieties, Load. Math. Soc. Lect. Notes Series 125 (1987)

44. P. Freyd, Abelian categories, Harper and Row (1964)

45. P. Freyd, A. Scedrov, Categories, allegories, North Holland (1990)

46. A. Fröhlich, Non-abelian homological algebra I, Derived functors and satellites, Proc. London Math. Soc. 11 (1961) 239-275

47. M. Gerstenhaber, A categorical setting for the Baer extension theory, Proc. in Symposia in Pure Mathematics 17 (1970) 50-64

48. A. W. Goldie, The Jordan-Hölder theorem for general abstract algebras, Proc. London Math. Math. Soc. 2 (1950) 107-113

49. M. Gran, Internal categories in Mal’cev categories, J. Pure Appl. Alg. 143 (1999) 221-229

50. M. Gran, Central extensions and internal groupolds in Maltsev categories, J. Pure Appl. Alg. 155 (2001) 139-166

51. M. Gran, Seni-abelian exact completions, Hornology, Homotopy and Appl. 4(1) (2002) 175-189

52. H. P. Gumm, Geometrical methods in congruence modular varieties, Mere. Amer. Math. Soc. 45 (1983)

53. J. Hagemann, C. Herrmann, A concrete ideal multiplication .for al- gebraic systems and its relation to congruence distributivity, Arch. Math. 32 (1979) 234-245

54. P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. 6 (1956) 366-416

55. P. J. Hilton and W. Ledermann, On the Jordan-Hölder theorem in homological monoids, Proc. London Math. Soc. 10 (1960) 321-334

56. F. Hofmann, Uber cine die Kategorie der Gruppen umfassende Kategorie, Sitzunss. Bayerische Akad. Wissensch. Math. Naturw. Klasse (1960) 163-204

57. S.A. Huq, Commutator, nilpotency and solvability in categories, Quart. J. Math. Oxford (2)19 (1968) 363-389

58. M. Huek, Productivity of properties of topological spaces, Topology Appl. 44 (1992) 189-196

59. G. Janelidze, Internal categories in Mal’cev varieties, Preprint York Univ. in Toronto (1990)

60. G. Janelidze, L. Márki, W. Tholen, Semi-abelian categories, J. Pure Appl. Alg. 168 (2002) 367-386

61. G. Janelidze, M. C. Pedicchio, Pseudogroupoids and commutator theory, Theory Appl. Categories 8 (2001) 405-456

62. P.T. Johnstone, Topos theory, London Math. Soc. Monographs 10, Aca- demic Press (1977)

63. P.T. Johnstone, Stone spaces, Cambridge Univ. Press (1982)

64. P. T. Johnstone, Affine categories and naturally Mal’cev categories, J. Pure Appl. Alg. 61 (1989) 251-256

65. P. T. Johnstone, The closed subgroup theorem for localic herds and pre- groupolds, J. Pure Appl. Alg. 70 (1991) 97-106

66. P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Vol. 1, 2, Oxford University Press, 2002

67. Peter Johnstone, A note on the semiabelian variety of Heyting semilattices, in: Galois theory, Hopf algebras, and semiabelian categories, 317–318, Fields Inst. Commun. 43, Amer. Math. Soc. 2004, MR2006a:18003

68. P. T. Johnstone and M. C. Pedicchio, Remarks on continuous Mal’cev algebras, Rend. Univ. Trieste (1995) 277-297

69. B. Jónsson and A. Tarski, Direct decompositions of finite algebraic systems, Notre Dame Mathematical Lectures, Notre Dame, Indiana (1947)

70. A. Kock, The algebraic theory of moving frames, Cahiers Top. Géom. Diff. Catégoriques 23 (1982) 347-362

71. A. Kock, Generalized fibre bundles, Springer Lect. Notes in Math. 1348 (1988) 194-207

72. A. Kock, Fibre bundle in general categories, J. Pure Appl. Alg. 56 (1989) 233–245

73. A. G. Kurosh, Direct decompositions in algebraic categories (Russian), Trudy Mosk. Mat. Obščestva. 8 (1959) 391-412

74. F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963) 869-873

75. F.W. Lawvere, R. Rosebrugh, Sets for mathematics, Cambridge University Press (2002)

76. F.W. Lawvere, S.H. Schanuel, Conceptual Mathematics, Buffalo Workshop preprint (1994)

77. F.E.J. Linton, An outline of functorial semantics, Spinger Lect. Notes Math. 80 (1969) 7–52

78. F.E.J. Linton, Applied functorial semantics, Springer LNM 80 (1969) 53-74

79. S. Mac Lane, Homology, Springer Verlag (1963)

80. S. Mac Lane, Categories for the working mathematician, 2nd edition, Springer Verlag (1998)

81. S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic, Universitext, Springer Verlag (1992)

82. A. I. Mal’cev, On the general theory of algebraic systems, Mat. Sbornik N. S. 35 (1954) 3-20

83. G. Orzech, Obstruction theory in algebraic categories, I and H, J. Pure Appl. Alg. 2 (1972) 287-340

84. M. C. Pedicchio, Maltsev categories and Maltsev operations, J. Pure Appl. Alg. 98 (1995) 67-71

85. M. C. Pedicchio, A categorical approach to commutator theory, J. Algebra 17’7’ (1995) 647-657

86. M. C. Pedicchio, Arithmetical categories and commutator theory, Appl. Categorical Structures 4 (1996) 297-305

87. R.S. Pierce, Modules over commutativc regular rings, Mere. Am. Math. Soc. 70 (1967)

88. A. F. Pixley, Distributivity and permutability of congruences in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963) 105-109

89. J. D. H. Smith, Mal’cev varieties, Springer Lect. Notes in Math. 554 (1976)

90. J. D. H. Smith, Centrality, Abstract of the Amer. Math. Soc. I (1980) 774-821

91. J. D. H. Smith, On the characterization of Maltsev and Jdnsson-Tarski algebras, Preprint Iowa State Univ. (2001)

92. M. S. Tsalenko, Correspondences over a quasi-exact category (Russian), Mat. Sbornik 73 (1967) 564-584

93. A. Ursini, On subtractive varieties, I, Algebra Universalis 31 (1994) 204-222

94. A. Ursini, On subtractive varieties, V: congruence modularity and commutators, Algebra Universalis 43 (2000) 51-78

95. V.V. Uspenskii, The Mal’tsev operation on countably compact spaces, Comment. mat. Univ. Carolinæ 30 (1989) 395-402

category: reference