Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-30T17:14:42.673Z Has data issue: false hasContentIssue false
  • English
  • Français

Groupoid Enriched Categories and Homotopy Theory

Published online by Cambridge University Press:  20 November 2018

P. H. H. Fantham  and
E. J. Moore
P. H. H. Fantham
Affiliation:
University of Toronto, Toronto, Ontario
E. J. Moore
Affiliation:
Scarborough College, University of Toronto, Scarborough, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We are concerned in this paper with category-theoretic aspects of homotopy theory. Originally, category theory developed as a simplifying language in the context of algebraic topology and yet one primary example: the category Π of spaces and homotopy classes of maps admits only limited use of the language owing to the very sparse occurrence of limits. Of course, full use has been made of them nevertheless: limits and colimits exist in the case of products and coproducts, and in almost no other case; yet, from this we obtain the theory of Samelson products, Whitehead products, and Hopf invariants which can all be expressed in Π see [8]. In addition, there are hosts of adjoint functors and yet the outcome is disappointing because the language applies only to special cases rather than to the situation as a whole.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 3 , 01 June 1983 , pp. 385 - 416
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Baumslag, G., Some aspects of groups with unique roots, Acta Math. 104 (1960), 217–303.Google Scholar
2. Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations. Lecture Notes in Math. 304 (Springer, Berlin-Heidelberg-New York, 1972).Google Scholar
3. Brown, E. H., Cohomology theories, Annals of Math. 75 (1962), 467–484; with a correction, Annals of Math. 78 (1963), 201.Google Scholar
4. Brown, E. H., Abstract homotopy theory, Trans. Am. Math. Soc. 119 (1965), 79–85.Google Scholar
5. Dold, A. and Thorn, R., Quasifaserungen unci unendliche symmetrische produkte, Annals of Math. 67 (1958), 239–281.Google Scholar
6. Ehresmann, C., Categories et structures, Travaux et Recherches Mathématiques 10 (Dunod, Paris, 1965).Google Scholar
7. Fantham, P. H. H., Lecture notes in homotopy theory (xerox notes, University of Toronto).Google Scholar
8. Fantham, P. H. H. and Moore, E.J., The homotopy theory of g.e. categories, in preparation.Google Scholar
9. Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory (Springer, Berlin-Heidelberg-New York, 1967).CrossRefGoogle Scholar
10. Gray, J. W., Formal category theory: adjointness for 2-categories, Lecture Notes in Math. 391 (Springer, Berlin-Heidelberg-New York, 1974).Google Scholar
11. Hilton, P. J., Mislin, G. and Roitberg, J., Localization theory for nilpotent groups and spaces, Mathematics Studies 15 (North Holland, Amsterdam-New York, 1975).Google Scholar
12. Kelly, G. M. and Street, R., Review of the elements of 2-categories, Lecture Notes in Math. 420 (Springer, Berlin-Heidelberg-New York, 1974), 75–103.Google Scholar
13. Massey, W. S., Algebraic topology: an introduction (Harcourt, Brace, and World, New York, 1967).Google Scholar
14. Mather, M., Pullbacks in homotopy theory, Can. J. Math. 28 (1976), 225–263.Google Scholar
15. Milnor, J., On spaces having the homotopy type of CW-complexes, Trans. Am. Math. Soc. 90 (1959), 272–280.Google Scholar
16. Mimura, M., Nashida, G. and Toda, H., Localization of CW complexes and applications, J. Math. Soc. Japan 23 (1971), 593–624.Google Scholar
17. Moore, E. J., Localization and an extended Brown representability theorem in a homotopy category, Ph. D. thesis, University of Toronto (1979).Google Scholar
18. Pareigis, B., Categories and functors (Academic Press, New York-London, 1970).Google Scholar
19. Quillen, D., Rational homotopy theory, Annals of Math. 90 (1969), 205–295.Google Scholar
20. Spencer, B., An abstract setting for homotopy pushouts and pullbacks, Cahiers de Topologie et Geom. Diff. 18 (1977), 409–429.Google Scholar
21. Street, R., Limits indexed by category valued 2-functors, J. Pure and Applied Algebra 8 (1976), 149–181.Google Scholar
22. Vogt, R. M., Homotopy limits and colimits. Math. Z. 134 (1973), 11–52.Google Scholar
23. Vogt, R. M., Commuting homotopy limits, Math. Z. 153 (1977), 59–82.Google Scholar
24. Walker, M., Homotopy pull-backs and applications to duality, Can. J. Math. 29 (1977), 45–64.Google Scholar