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Homotopy of Natural Transformations

Published online by Cambridge University Press:  20 November 2018

K. A. Hardie
K. A. Hardie*
Affiliation:
University of Cape Town, Rondebosch, C. P., Republic of South Africa
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Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphisms

such that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 332 - 341
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Hardie, K. A., Weak homotopy equivalence of P-functors, Quart. J. Math. Oxford Ser. (2) 19 (1968), 17–31.Google Scholar
2. Holm, P., Excision and cofibrations, Math. Scand. 19 (1966), 122–126.Google Scholar
3. Michael, E., espaces, J. Math. Mech. 15 (1966), 983–1002.Google Scholar
4. Milnor, J., The construction FK, Princeton University, mimeographed notes, 1956.Google Scholar
5. Porter, G. J., Higher order Whitehead products, Doctoral dissertation, Cornell University, Ithaca, New York, 1963.Google Scholar
6. Porter, G. J., Higher order Whitehead products, Topology 3 (1965), 123–135.Google Scholar
7. Puppe, D., Homotopiemengen und ihre induzierten Abbildungen. I, Math. Z. 69 (1958), 299–344.Google Scholar
8. Puppe, D., Bemerkungen über die Erweiterung von Homotopien, Arch. Math. 18 (1967), 81–88.Google Scholar
9. Spanier, E. H., Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar
10. Strom, A., Note on cofibrations, Math. Scand. 19 (1966), 11–14.Google Scholar