On the categorical foundations of homological and homotopical algebra

Marco Grandis

Cahiers de Topologie et Géométrie Différentielle Catégoriques (1992)

  • Volume: 33, Issue: 2, page 135-175
  • ISSN: 1245-530X

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Grandis, Marco. "On the categorical foundations of homological and homotopical algebra." Cahiers de Topologie et Géométrie Différentielle Catégoriques 33.2 (1992): 135-175. <http://eudml.org/doc/91493>.

@article{Grandis1992,
author = {Grandis, Marco},
journal = {Cahiers de Topologie et Géométrie Différentielle Catégoriques},
keywords = {homotopical algebra; homological algebra; exact categories; homotopy pushout},
language = {eng},
number = {2},
pages = {135-175},
publisher = {Dunod éditeur, publié avec le concours du CNRS},
title = {On the categorical foundations of homological and homotopical algebra},
url = {http://eudml.org/doc/91493},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Grandis, Marco
TI - On the categorical foundations of homological and homotopical algebra
JO - Cahiers de Topologie et Géométrie Différentielle Catégoriques
PY - 1992
PB - Dunod éditeur, publié avec le concours du CNRS
VL - 33
IS - 2
SP - 135
EP - 175
LA - eng
KW - homotopical algebra; homological algebra; exact categories; homotopy pushout
UR - http://eudml.org/doc/91493
ER -

References

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  1. [An] D.W. Anderson, Axiomatic homotopy theory, in: Algebraic Topology, Waterloo, 1978, p. 520-547, Lect. Notes in Math.741, Springer1979. Zbl0438.55010MR557184
  2. [Ba] H.J. Baues, Algebraic homotopy, Cambridge Univ. Press1989. Zbl0688.55001MR985099
  3. [BK] A.K. Bousfield - D.M. Kan, Homotopy limits, completions and localizations, Lect. Notes in Math.304, Springer1972. Zbl0259.55004MR365573
  4. [Br] K.S. Brown, Abstract homotopy theory and generalised sheaf cohomology, Trans. Amer. Math. Soc.186 (1973), 419-458. Zbl0245.55007MR341469
  5. [Bu] D.A. Buchsbaum, Exact categories and duality, Trans. Amer. Math. Soc.80 (1955), 1-34. Zbl0065.25502MR74407
  6. [CE] H. Cartan - S. Eilenberg, Homological algebra, Princeton Univ. Press1956. Zbl0075.24305MR77480
  7. [DT] A. Dold - R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math.67 (1958), 239-281. Zbl0091.37102MR97062
  8. [Eh] C. Ehresmann, Cohomologie à valeurs dans une catégorie dominée, Extraits du Colloque de Topologie, Bruxelles1964, in: C. Ehresmann, Oeuvres complètes et commentées, Partie III-2, 531-590, Amiens1980 (8). Zbl0201.35502
  9. [ES] S. Eilenberg - N. Steenrod, Foundations of algebraic topology, Princeton Univ. Press1952. Zbl0047.41402MR50886
  10. [G1] M. Grandis, Symétrisations de categories et catégories quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. 14 sez. 1 (1977), 133-207. Zbl0418.18003MR472959
  11. [G2] M. Grandis, On distributive homological algebra, I-III. Cahiers Top. Géom. Diff.25 (1984), 259-301; 353-379; 26 (1985), 169-213. Zbl0564.18005
  12. [G3] M. Grandis, A categorical approach to exactness in algebraic topology, in: Atti del V Convegno Intemazionale di Topologia, Lecce-Otranto1990, Rend. Circ. Mat. Palermo, to appear. Zbl0770.55013MR1197170
  13. [G4] M. Grandis, On homological algebra, I. Semiexact and homological categories, Dip. Mat. Univ. Genova, Preprint 186 (1991), 1-45 (9). 
  14. [G5] M. Grandis, On homological algebra, II. Homology and satellites, Dip. Mat. Univ. Genova, Preprint 187 (1991), 1-39. 
  15. [G6] M. Grandis, Homotopical algebra: a two-dimensional categorical setting, Dip. Mat. Univ. Genova, Preprint 191 (1991), 1-50. 
  16. [Ga] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France90 (1962), 323-348. Zbl0201.35602MR232821
  17. [Gr] A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J.9 (1957), 119-221. Zbl0118.26104MR102537
  18. [GV1] A.R. Grandjeán - L. Valcárcel, Homología en categorías exactas, Dep. de Algebra y Fundamentos, Santiago de Compostela, 1970. Zbl0291.18014MR285586
  19. [GV2] A.R. Grandjeán - L. Valcárcel, Pares exactos y sucesiones espectrales, ibidem, 1974. 
  20. [GZ] P. Gabriel - M. Zisman, Calculus of fractions and homotopy theory, Springer1967. Zbl0186.56802MR210125
  21. [H1] A. Heller, Stable homotopy categories, Bull. Amer. Math. Soc.74 (1968), 28-63. Zbl0177.25605MR224090
  22. [H2] A. Heller, Homotopy theories, Mem. Amer. Math. Soc.383 (1988). Zbl0643.55015MR920963
  23. [Hb1] P.J. Huber, Homotopy theories in general categories, Math. Ann.144 (1961), 361-385. Zbl0099.17905MR150184
  24. [Hb2] P.J. Huber, Standard constructions in abelian categories, Math. Ann.146 (1962), 321-325. Zbl0101.40701MR150185
  25. [K1] K.H. Kamps, Über einige formale Eigenschaften von Faserungen und h-Faserungen, Manuscripta Math.3 (1970), 237-255. Zbl0205.27502MR270374
  26. [K2] K.H. Kamps, Zur Homotopietheorie von Gruppoiden, Arch. Math.(Basel) 23 (1972), 610-618. Zbl0251.20059MR315022
  27. [Ka] D.M. Kan, Abstract homotopy I-IV, Proc. Nat. Acad. Sci. U.S.A.41 (1955), 1092-1096; 42 (1956), 255-258; 419-421; 542-544. Zbl0065.38601MR79762
  28. [Ke] G.M. Kelly, Single-space axioms for homology theories, Proc. Cambridge Philos. Soc.55 (1959), 10-22. Zbl0084.19001MR108787
  29. [Ks] H. Kleisli, Homotopy theory in abelian categories, Canad. J. Math.14 (1962), 139-169. Zbl0108.02001MR141696
  30. [La] R. Lavendhomme, Un plongement pleinement fidèle de la catégorie des groupes, Bull. Soc. Math. Belgique, 17 (1965), 153-185. Zbl0146.02602MR186713
  31. [LV] R. Lavendhomme - G. Van Den Bossche, Homologisation de catégories prerégulières, Inst. de Math. Pure Appl., Univ. Catholique de Louvain41 (1973), 1-24. 
  32. [Ma] S. Mac Lane, Homology, Springer, 1963. Zbl0818.18001MR1344215
  33. [Mc] H.J. Marcum, Homotopy equivalences in 2-categories, in: Groups of self-equivalences and related topics, Montréal1988, p. 71-86, Lect. Notes in Math.1425, Springer1990. Zbl0712.55006MR1070577
  34. [Mh] M. Mather, Pull-backs in homotopy theory, Can. J. Math.28 (1976), 225-263. [Mt] B. Mitchell, Theory of categories, Academic Press1965. Zbl0351.55005MR402694
  35. [P1] D. Puppe, Homotopiemengen und ihre induzierten Abbildungen, I, Math. Z.69 (1958), 299-344. Zbl0092.39803MR100265
  36. [P2] D. Puppe, Korrespondenzen in abelschen Kategorien, Math. Ann.148 (1962), 1-30. Zbl0109.25201MR141698
  37. [Qn] D.G. Quillen, Homotopical algebra, Lect. Notes in Math.43, Springer1967. [Ze] E.C. Zeeman, On the filtered differential group, Ann. Math.66 (1957), 557-585. Zbl0168.20903MR223432

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