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Vogt's theorem on categories of homotopy coherent diagrams

Published online by Cambridge University Press:  24 October 2008

Jean-Marc Cordier  and
Timothy Porter
Jean-Marc Cordier
Affiliation:
U.E.R. de Mathématiques, Université de Picardie, 80039 Amiens, France
Timothy Porter
Affiliation:
Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd LL57 2UW
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Extract

Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the category of simplicial sets). For A a small category, Vogt (in [22]) constructed a category, Coh (A, Top), of homotopy coherent A-indexed diagrams in Top and homotopy classes of homotopy coherent maps, and proved a theorem identifying this as being equivalent to Ho (TopA), the category obtained from the category of commutative A-indexed diagrams by localizing with respect to the level homotopy equivalences. Thus one of the important consequences of Vogt's result is that it provides concrete coherent models for the formal composites of maps and formal inverses of level homotopy equivalences which are the maps in Ho (TopA). The usefulness of such models and in general of Vogt's results is shown in the series of notes [14–17] by the second author in which those results are applied to give an obstruction theory applicable in prohomotopy theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 1 , July 1986 , pp. 65 - 90
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Boardman, J. M. and Vogt, R. M.. Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Math. vol. 347 (Springer, 1973).Google Scholar
[2]Borceux, F. and Kelly, G. M.. A notion of limit for enriched categories. Bull. Austral. Math. Soc. 12 (1975), 49–72.CrossRefGoogle Scholar
[3]Bourn, D. and Cordier, J.-M.. A general formulation of homotopy limits. J. Pure Appl. Algebra 29 (1983), 129–141.CrossRefGoogle Scholar
[4]Bousfield, A. K. and Kan, D. M.. Homotopy Limits, Completions and Localizations. Lecture Notes in Math. vol. 304 (Springer, 1972).Google Scholar
[5]Bozapalides, S.. Some remarks on Lax-presheafs. Illinois J. Math. 24 (1980), 676–680.CrossRefGoogle Scholar
[6]Cordier, J. -M.. Sur la notion de diagramme homotopiquement cohérent, Proc 3éme Colloque sur les Catégories, Amiens (1980). Cahiers Topologie Géom. Différentielle 23 (1982), 93–112.Google Scholar
[7]Cordier, J.-M. and Porter, T.. Homotopy Limits and Homotopy Coherence. Lectures given at Perugia (Sept.–Oct., 1984), (in preparation).Google Scholar
[8]Edwards, D. A. and Hastings, H. M., Čech and Steenrod Homotopy Theories with Applications to Geometric Topology. Lecture Notes in Math. vol. 542 (Springer, 1976).CrossRefGoogle Scholar
[9]Gabriel, P. and Zisman, M.. Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik, vol. 35 (Springer, 1967).CrossRefGoogle Scholar
[10]Giraud, J.. Méthode de la descente, Bulletin Soc. Math. Soc. France Memoire 2, 1964.Google Scholar
[11]Kelly, G. M.. The Basic Concepts of Enriched Category Theory, LMS Lecture Notes no. 64 (Cambridge University Press, 1983).Google Scholar
[12]Lashof, R.. The immersion approach to triangulation and smoothing. Proc. Adv. Study Inst. on Alg. Top, Math. Inst. (Aarhus 1970).CrossRefGoogle Scholar
[13]Mac Lane, S.. Categories for the Working Mathematician, Grad. Texts in Math. 5 (Springer, 1971).CrossRefGoogle Scholar
[14]Porter, T.. Stability results for topological spaces. Math. Z. 140 (1974), 1–21.Google Scholar
[15]Porter, T.. Abstract homotopy theory in procategories. Cahiers Topologie Géom. Différentielle 17 (1976), 113–124.Google Scholar
[16]Porter, T.. Coherent prohomotopical algebra. Cahiers Topologie Géom. Différentielle 18 (1977), 139–179.Google Scholar
[17]Porter, T.. Coherent prohomotopy theory. Cahiers Topologie Géom. Différentielle 19 (1978), 3–44.Google Scholar
[18]Segal, G. B.. Categories and cohomology theories. Topology 13 (1974), 293–312.Google Scholar
[19]Street, R.. Two constructions on lax functors. Cahier Topologie Géom. Différentielle 13 (1972), 217–264.Google Scholar
[20]Spencer, C. B.. An abstract setting for homotopy pushouts and pullbacks. Cahiers Topologie Géom. Différentielle 18 (1977), 409–429.Google Scholar
[21]Vogt, R. M.. A note on homotopy equivalences. Proc. Amer. Math. Soc. 32 (1972), 627–629.Google Scholar
[22]Vogt, R. M.. Homotopy limits and colimits. Math. Z. 134 (1973), 11–52.CrossRefGoogle Scholar
[23]Dwyer, W. G. and Kan, D. M.. Equivalences between homotopy theories of diagrams. (Preprint 1984).Google Scholar
[24]Heller, A.. Homotopy in functor categories. Trans. Amer. Math. Soc. 272 (1982), 185–202.CrossRefGoogle Scholar