Categorical strong shape theory

Mikhail A. Batanin

Cahiers de Topologie et Géométrie Différentielle Catégoriques (1997)

  • Volume: 38, Issue: 1, page 3-66
  • ISSN: 1245-530X

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Batanin, Mikhail A.. "Categorical strong shape theory." Cahiers de Topologie et Géométrie Différentielle Catégoriques 38.1 (1997): 3-66. <http://eudml.org/doc/91586>.

@article{Batanin1997,
author = {Batanin, Mikhail A.},
journal = {Cahiers de Topologie et Géométrie Différentielle Catégoriques},
keywords = {shape theory; strong shape theory; compact metric spaces; categories of inverse systems; compact Hausdorff spaces; Kan extensions; shape category; Kleisli category; homotopy category of polyhedra; homotopy coherence problems; homotopical algebra; geometric topology; coherent prohomotopy},
language = {eng},
number = {1},
pages = {3-66},
publisher = {Dunod éditeur, publié avec le concours du CNRS},
title = {Categorical strong shape theory},
url = {http://eudml.org/doc/91586},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Batanin, Mikhail A.
TI - Categorical strong shape theory
JO - Cahiers de Topologie et Géométrie Différentielle Catégoriques
PY - 1997
PB - Dunod éditeur, publié avec le concours du CNRS
VL - 38
IS - 1
SP - 3
EP - 66
LA - eng
KW - shape theory; strong shape theory; compact metric spaces; categories of inverse systems; compact Hausdorff spaces; Kan extensions; shape category; Kleisli category; homotopy category of polyhedra; homotopy coherence problems; homotopical algebra; geometric topology; coherent prohomotopy
UR - http://eudml.org/doc/91586
ER -

References

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  1. [1] Artin M., Mazur B., On the van Kampen Theorem, Topology, 5, pp.179-189, 1966. Zbl0138.18301MR192495
  2. [2] Batanin M.A., Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Topologie et Geom. Diff., vol.XXXIV-4, pp.279-304, 1993. Zbl0793.18005MR1253172
  3. [3] Batanin M.A., Homotopy coherent category theory and A∞structures in monoidal categories, to appear. Zbl0892.18003
  4. [4] Bauer F.W., A shape theory with singular homology, Pacific J. Math., 64, pp.25-65, 1976. Zbl0346.55014MR436130
  5. [5] Benabou J., Introduction to bicategories, in Lecture Notes in Math., vol.47, pp.1-77Springer-Verlag, Berlin, Heidelberg, New York, 1967. MR220789
  6. [6] Benabou J., Les distributeurs, Inst. Math. Pures and Appl., Univ. Louvain-la-Neuve, Rapport No.33, 1973. 
  7. [7] Boardman J.M., R.M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Math., vol. 347, Springer-Verlag, Berlin, Heidelberg, New York, 1973. Zbl0285.55012MR420609
  8. [8] Bourn D., Cordier J.-M., Distributeurs et theorie de la forme, Cahiers Topologie et Geom. Diff., 21, pp.161-189, 1980. Zbl0439.55014MR574663
  9. [9] Bourn D., Cordier J.-M., A general formulation of homotopy limits, J.Pure Appl. Algebra, 29, pp.129-141, 1983. Zbl0575.55006MR707615
  10. [10] Bousfield A.K., Kan D.M., Homotopy Limits, Completions and Localizations, Lecture Notes in Math., vol.304, Springer-Verlag, Berlin, Heidelberg, New York, 1972. Zbl0259.55004MR365573
  11. [11] Cathey F.W., Segal J., Strong shape theory and resolutions, Topology Appl.15, pp.119-130, 1983. Zbl0505.55012MR686090
  12. [12] Cathey F.W., Segal J., Homotopical approach to strong shape or completion theory, Topology Appl.21, pp.167-192, 1985. Zbl0588.55008MR813287
  13. [13] Cordier J.-M., Extension de Kan simplicialement coherent, Pre-publication, Amiens, 1985. 
  14. [14] Cordier J.-M., Comparaison de deux catégories d'homotopie de morphismes cohérents, Cahiers de Topologie et Géometrie Diff. Catégoriques, vol. 30-2, pp. 257-275, 1989. Zbl0679.55006MR1029628
  15. [15] Cordier J.-M., Porter T., Vogt's Theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Phil. Soc., vol. 100, pp 65-90, 1986. Zbl0603.55017MR838654
  16. [16] Cordier J.-M., Porter T., Shape Theory: Categorical Methods of Approximation, Ellis Horwood Limited, Chichester1989. Zbl0663.18001MR1000348
  17. [17] Cordier J.-M., Porter T., Maps between homotopy coherent diagrams, Topology and its Appl., 28, pp.255-275, 1988. Zbl0655.55008MR931527
  18. [18] Cordier J.-M., Porter T., Fibrant diagrams, rectifications and a construction of Loday, J.Pure Appl.Algebra, 67, pp.111-124, 1990. Zbl0715.55012MR1080880
  19. [19] Cordier J.-M., Porter T., Homotopy coherent category theory, to appear in Transactions of the AMS, Zbl0865.18006MR1376543
  20. [20] Dwyer W.G., Kan D.M., Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc., 91, pp.456-460, 1984. Zbl0514.55020MR744648
  21. [21] Dwyer W.G., Kan D.M., Function complexes in homotopical algebra, Topology19, pp.427-440, 1980. Zbl0438.55011MR584566
  22. [22] Dydak J., Nowak S., Strong shape for topological spaces, Transactions of the AMS, 323, n.2, pp.765-796, 1991. Zbl0754.55009MR986690
  23. [23] Edwards D.A., Hastings H.M., Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Math.542, Springer-Verlag, Berlin, 1976. Zbl0334.55001MR428322
  24. [24] Gabriel P., Zisman M., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 35, Springer, 1967. Zbl0186.56802MR210125
  25. [25] Grothendieck A., Pursuing Stacks, typed manuscript, (600 pp.), 1983. 
  26. [26] Günther B., The use of semisimplicial complexes in strong shape theory, Glasnik Mat., 27, pp.101-144, 1992. Zbl0781.55006MR1230117
  27. [27] Günther B., Comparison of the Coherent pro-Homotopy Theories of Edwards-Hastings, Lisica-Mardešić and Günther, Glasnik Mat., 26, pp.141-176, 1991. Zbl0795.55010MR1269184
  28. [28] Heller A., Homotopy in Functor Categories, Transactions AMS., v272.272, pp.185-202, 1982. Zbl0508.55022MR656485
  29. [29] Heller A., Homotopies theories, Memoirs AMS., v.71, 383, AMS, Rhode Island, USA, 1988. Zbl0643.55015MR920963
  30. [30] Kelly G.M., The Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series, 64, (Cambridge University Press, Cambridge), 1983. Zbl0478.18005MR651714
  31. [31] Lada T.J., Strong homotopy algebras over monads, in Lecture Notes in Math., vol. 533, pp.399-479Springer-Verlag, Berlin, Heidelberg, New York, 1976. 
  32. [32] Lisica Y.T., Mardešić S., Coherent prohomotopy and strong shape theory, Glasnik Mat., 19, pp.335-399, 1984. Zbl0553.55009MR790021
  33. [33] MacLane S., Categories for the working mathematician, Graduate Texts in Math.5, Springer-Verlag, Berlin, Heidelberg, New York, 1971. Zbl0232.18001MR354798
  34. [34] MacLane S., Paré R., Coherence for bicategories and indexed categories, J. Pure Appl. Algebra, 37, pp.59-80, 1985. Zbl0567.18003MR794793
  35. [35] Mardešić S., Strong expansions and strong shape theory, Topology Appl., 38, pp. 275-292, 1991. Zbl0715.55008MR1098907
  36. [36] Mardešić S., Segal J., Shape theory, North-Holland, Amsterdam, 1982. Zbl0495.55001MR676973
  37. [37] May J.P., The Geometry of Iterated Loop Spaces, Lecture Notes in Math., vol. 271, Springer-Verlag, Berlin, Heidelberg, New York, 1972. Zbl0244.55009MR420610
  38. [38] Meyer J.-P., Bar and cobar constructions, J. Pure Applied Alg., 33, pp.163-207, 1984. Zbl0542.57036MR754954
  39. [39] Porter T., Stability results for topological spaces, Math. Z., 140, pp. 1-21, 1974. Zbl0275.55022MR385846
  40. [40] Porter T., On the two definitions of Ho(pro — C), Topology Appl., 28, pp. 289-293, 1988. Zbl0647.18008MR931529
  41. [41] Quillen D.G., Homotopical Algebra, Lecture Notes in Math., vol. 43, Springer-Verlag, Berlin, Heidelberg, New York, 1967. Zbl0168.20903MR223432
  42. [42] Schwänzl R., Vogt R., Homotopy homomorphisms and the Hammock localization, Boletin de la Soc. Mat. Mexicana, 37, 1-2, pp.431-449, 1992. Zbl0853.55010MR1317592
  43. [43] Smirnov V.A., Homotopy theory of coalgebras, Math. USSR Izv., 27, pp.575-592, 1986. Zbl0612.55012MR816858
  44. [44] Vogt R.M., Homotopy limits and colimits, Math. Z, 134, pp.11-52, 1973. Zbl0276.55006MR331376

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