Theory and Applications of Categories, Vol. 23, 2010, pp. 1–6.

FOREWORD

This special volume of Theory and Applications of Categories is dedicated to our colleague Professor Dominique Bourn on the occasion of his 60th birthday and in appreciation of his remarkable contribution to category theory and categorical algebra.^{1}

Dominique has prepared his “thèse de 3ème cycle” (1973) on internal monads. He obtained his “Doctorat d’Etat” in 1990 in Paris XIII, with Jean Bénabou as Director; his thesis consists of four main papers on cohomology. Since the beginning of his research activity, Dominique has been actively involved in the stimulating environment of the “Theory and Applications of Categories” group active in Paris and Amiens initiated by Charles Ehresmann and Andrée (Bastiani-)Ehresmann. He was Assistant in Lille from 1973 to 1976 and “Maître de Conférences” in Amiens from 1977 to 1993, where, beginning from 1984, he was in charge of the direction of the TAC research team. Since 1993, he is Professor at the Université du Littoral in Calais-Dunkerque. At this young university, he created the pure mathematics laboratory, organized several SIC (Séminaire Itinérant de Catégories) and PSSL (Peripatetic Seminar on Sheaves and Logic) meetings and, in June 2008, the International Category Theory Conference.

We want to express our admiration to Dominique for his beautiful and highly original ideas and results, including his approach to cohomology theories, higher-dimensional category theory and n-categorification, his concept of protomodularity, a cascade of other new concepts and results involving what he calls the fibration of points, non-abelian versions of classical homological lemmas in homological categories, and many more expressed in his 76 publications listed below.

We thank the editors of Theory and Applications of Categories, and especially the Managing Editor Bob Rosebrugh, for the opportunity to prepare this volume.

The Guest Editors: A.C. Ehresmann, G. Janelidze, J. Rosický, W. Tholen and E.M. Vitale

Published on 2010-01-25.

\begin{matrix}c \\ ◯ \end{matrix} Editors of Theory and Applications of Categories, 2010. Permission to copy for private use granted.

1The present volume of TAC follows an International Conference organized in October 2008 at the Royal Flemish Academy of Belgium in Brussels to celebrate Dominique Bourn and Francis Borceux 60th birthday. A special volume of the Cahiers de Topologie et Géom. Diff. Catégoriques is devoted to Francis Borceux.

2  FOREWORD

Publications of Professor Dominique Bourn

1. Triples p-structurés. Cahiers Topologie Géom. Différentielle 13 (1972), 327–356.

2. Triples p-structurés. C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A703–A705.

3. Sur les triples structurés. C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A255–A258.

4. Sur les objets de Kleisli et d’Eilenberg-Moore dans une 2-catégorie. C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1041–A1044.

5. Anadèses et catadèses naturelles. C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1401–A1404.

6. Anadèses et catadèses naturelles entre 3-foncteurs. C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A1025–A1028.

7. Natural anadeses and catadeses. Conférences du Colloque sur l’Algèbre des Catégories (Amiens, 1973), I. Cahiers Topologie Géom. Différentielle 14 (1973), no. 4, 371–415, 480.

8. Objets de Kleisli et d’Eilenberg-Moore d’un triple dans une 2-catégorie. Thèse 3ème cycle, Specialité: Mathématiques, Université de Paris VII. Esquisses mathématiques, No. 19, Exp. No. 1, iv+63 pp. Fac. Sci. Univ. Paris VII, Paris, 1973.

9. Analgèbres et catalgèbres. C. R. Acad. Sci. Paris Sér. A 278 (1974), 317–320.

10. Ditopos. C. R. Acad. Sci. Paris Sér. A 279 (1974), 731–732.

11. Sur les ditopos. C. R. Acad. Sci. Paris Sér. A 279 (1974), 911–913.

12. A propos des ditopos. Nordwestdeutsches Kategorienseminar (Tagung, Bremen, 1976), pp. 11–25. Math.-Arbeitspapiere, No. 7, Teil A: Math. Forschungspapiere, Univ. Bremen, Bremen, 1976.

13. (with Penon, Jacques). 2-catégories réductibles. U.E.R Math., Univ. Picardie, Amiens (1978).

14. (with Cordier, Jean-Marc). Distributeurs et théorie de la forme. Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 161–189.

15. Une autre propriété universelle pour le champ associé. Third Colloquium on Categories (Amiens, 1980), Part I. Cahiers Topologie Géom. Différentielle 21 (1980), no. 4, 403–410.

16. A canonical action on indexed limits: an application to coherent homotopy. Category theory (Gummersbach, 1981), pp. 23–32, Lecture Notes in Math., 962, Springer, Berlin-New York, 1982.

17. (with Cordier, Jean-Marc). A general formulation of homotopy limits. J. Pure Appl. Algebra 29 (1983), no. 2, 129–141.

18. La tour de fibrations exactes des n-catégories. Cahiers Topologie Géom. Différentielle Catég. 25 (1984), no. 4, 327–351.

19. Une théorie de cohomologie pour les catégories exactes. C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 5, 173–176.

20. The shift functor and the comprehensive factorization for internal groupoids. Cahiers Topologie Géom. Différentielle Catég. 28 (1987), no. 3, 197–226.

21. Pseudofunctors and nonabelian weak equivalences. Categorical algebra and its applications (Louvain-La-Neuve, 1987), 55–71, Lecture Notes in Math., 1348, Springer, Berlin, 1988.

22. A right exactness property for internal categories. Cahiers Topologie Géom. Différentielle Catég. 29 (1988), no. 2, 109–155.

23. Produits tensoriels cohérents de complexes de chaîne. Actes du Colloque en l’Honneur du Soixantième Anniversaire de René Lavendhomme (Louvain-la-Neuve, 1989). Bull. Soc. Math. Belg. Sér. A 41 (1989), no. 2, 219–247.

24. The tower of n-groupoids and the long cohomology sequence. J. Pure Appl. Algebra 62 (1989), no. 2, 137–183.

25. Another denormalization theorem for abelian chain complexes. J. Pure Appl. Algebra 66 (1990), no. 3, 229–249.

26. Normalization equivalence, kernel equivalence and affine categories. Category theory (Como, 1990), 43–62, Lecture Notes in Math., 1488, Springer, Berlin, 1991.

27. Low-dimensional geometry of the notion of choice. Category theory 1991 (Montreal, PQ, 1991), 55–73, CMS Conf. Proc., 13, Amer. Math. Soc., Providence, RI, 1992.

28. Note on a submonadicity. Cahiers Topologie Géom. Différentielle Catég. 33 (1992), no. 3, 199–206.

29. Polyhedral monadicity of n-groupoids and standardized adjunction. J. Pure Appl. Algebra 99 (1995), no. 2, 135–181.

30. Mal’cev categories and fibration of pointed objects. The European Colloquium of Category Theory (Tours, 1994). Appl. Categ. Structures 4 (1996), no. 2-3, 307– 327.

31. (with Janelidze, George). Protomodularity, descent, and semidirect products. Theory Appl. Categ. 4 (1998), No. 2, 37–46 .

32. Baer sums and fibered aspects of Mal’cev operations. Cahiers Topologie Géom. Différentielle Catég. 40 (1999), no. 4, 297–316.

33. The structural nature of the nerve functor for n-groupoids. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures 8 (2000), no. 1-2, 81–113.

34. Normal functors and strong protomodularity. Theory Appl. Categ. 7 (2000), No. 9, 206–218 .

35. Normal subobjects and abelian objects in protomodular categories. J. Algebra 228 (2000), no. 1, 143–164.

36. A categorical genealogy for the congruence distributive property. Theory Appl. Categ. 8 (2001), No. 14, 391–407 .

37. 3 × 3 lemma and protomodularity. J. Algebra 236 (2001), no. 2, 778–795.

38. n-groupoids from n-truncated simplicial objects as a solution to a universal problem. Category theory and its applications (Montreal, QC, 1997). J. Pure Appl. Algebra 154 (2000), no. 1-3, 71–102.

39. (with Gran, Marino). Centrality and normality in protomodular categories. CT2000 Conference (Como). Theory Appl. Categ. 9 (2001/02), 151–165 .

40. (with Vitale, Enrico). Extensions of symmetric cat-groups. Homology Homotopy Appl. 4 (2002), no. 1, 103–162 .

41. (with Gran, Marino). Centrality and connectors in Maltsev categories. Algebra Universalis 48 (2002), no. 3, 309–331.

42. Aspherical abelian groupoids and their directions. Category theory 1999 (Coimbra). J. Pure Appl. Algebra 168 (2002), no. 2-3, 133–146.

43. (with Gran, Marino). Central extensions in semi-abelian categories. Special volume celebrating the 70th birthday of Professor Max Kelly. J. Pure Appl. Algebra 175 (2002), no. 1-3, 31–44.

44. Intrinsic centrality and associated classifying properties. J. Algebra 256 (2002), no. 1, 126–145.

45. (with Janelidze, George). Characterization of protomodular varieties of universal algebras. Theory Appl. Categ. 11 (2003), No. 6, 143–147 .

46. The denormalized 3 × 3 lemma. J. Pure Appl. Algebra 177 (2003), no. 2, 113–129.

47. Commutator theory in strongly protomodular categories. Theory Appl. Categ. 13 (2004), No. 2, 27–40 .

48. (with Janelidze, George). Extensions with abelian kernels in protomodular categories. Georgian Math. J. 11 (2004), no. 4, 645–654.

49. (with Gran, Marino). Normal sections and direct product decompositions. Comm. Algebra 32 (2004), no. 10, 3825–3842.

50. Commutator theory in regular Mal’cev categories. Galois theory, Hopf algebras, and semiabelian categories, 61–75, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004.

51. (with Gran, Marino). Categorical aspects of modularity. Galois theory, Hopf algebras, and semiabelian categories, 77–100, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004.

52. Protomodular aspect of the dual of a topos. Adv. Math. 187 (2004), no. 1, 240–255.

53. (with Gran, Marino). Regular, protomodular, and abelian categories. Categorical foundations, 165–211, Encyclopedia Math. Appl., 97, Cambridge Univ. Press, Cambridge, 2004.

54. (with Borceux, Francis). Mal’cev, protomodular, homological and semi-abelian categories. Mathematics and its Applications, 566. Kluwer Academic Publishers, Dordrecht, 2004. xiv+479 pp.

55. Congruence distributivity in Goursat and Mal’cev categories. Appl. Categ. Structures 13 (2005), no. 2, 101–111.

56. (with Penon, Jacques). Catégorification de structures définies par monade cartésienne. Cah. Topol. Géom. Différ. Catég. 46 (2005), no. 1, 2–52.

57. On the direct image of intersections in exact homological categories. J. Pure Appl. Algebra 196 (2005), no. 1, 39–52.

58. Fibration of points and congruence modularity. Algebra Universalis 52 (2004), no. 4, 403–429 (2005).

59. (with Borceux, Francis and Johnstone, Peter). Initial normal covers in bi-Heyting toposes. Arch. Math. (Brno) 42 (2006), no. 4, 335–356.

60. (with Gran, Marino). Torsion theories in homological categories. J. Algebra 305 (2006), no. 1, 18–47.

61. Action groupoid in protomodular categories. Theory Appl. Categ. 16 (2006), No. 2, 46–58.

62. Normal subobjects of topological groups and of topological semi-abelian algebras. Topology Appl. 153 (2006), no. 8, 1341–1364.

63. Distributive law, commutator theory and Yang-Baxter equation. JP J. Algebra Number Theory Appl. 8 (2007), no. 2, 145–163.

64. (with Borceux, Francis). Split extension classifier and centrality. Categories in algebra, geometry and mathematical physics, 85–104, Contemp. Math., 431, Amer. Math. Soc., Providence, RI, 2007.

65. Moore normalization and Dold-Kan theorem for semi-abelian categories. Categories in algebra, geometry and mathematical physics, 105–124, Contemp. Math., 431, Amer. Math. Soc., Providence, RI, 2007.

66. (with Rodelo, Diana). Cohomology without projectives. Cah. Topol. Géom. Différ. Catég. 48 (2007), no. 2, 104–153.

67. Baer sums in homological categories. J. Algebra 308 (2007), no. 1, 414–443.

68. The Mal’tsev operation on extensions. Cah. Topol. Géom. Différ. Catég. 49 (2008), no. 3, 176–195.

69. Abelian groupoids and non-pointed additive categories. Theory Appl. Categ. 20 (2008), No. 4, 48–73.

70. Commutator theory, action groupoids, and an intrinsic Schreier-Mac Lane extension theorem. Adv. Math. 217 (2008), no. 6, 2700–2735.

71. (with Rosický, Jirí). Mal’cev conditions revisited. Appl. Categ. Structures 16 (2008), no. 6, 723–733.

72. (with Janelidze, Zurab). Approximate Mal’tsev operations. Theory Appl. Categ. 21 (2008), No. 8, 152–171.

73. (with Janelidze, Zurab). Pointed protomodularity via natural imaginary subtractions. J. Pure Appl. Algebra 213 (2009), no. 9, 1835–1851.

74. (with Janelidze, Zurab). Subtractive categories and extended subtractions. Appl. Categ. Structures 17 (2009), no. 4, 317–343.

75. (with Janelidze, George). Centralizers in action accessible categories. Cah. Topol. Géom. Différ. Catég. 50 (2009), no. 3, 211–232.

76. The cohomological comparison arising from the associated abelian object. Available as arXiv:1001.0905 (2010), 24 pp.

This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/fwd/23-fwd.{dvi,ps,pdf}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, McGill University: gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Cuill Inc.: valeria@cuill.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Tom Leinster, University of Glasgow, T.Leinster@maths.gla.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca