An idempotent completion functor in homotopy theory
Author:
Harold M. Hastings
Journal:
Trans. Amer. Math. Soc. 287 (1985), 387402
MSC:
Primary 55P60; Secondary 55U35
MathSciNet review:
766226
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Abstract: We observe that ArtinMazur style completions ( is a commutative ring with identity) induce analogous idempotent completions on the weak prohomotopy category proHo(Top). Because Ho(Top) is a subcategory of proHo(Top) and proHo(Top) is closely related to the topologized homotopy category of J. F. Adams and D. Sullivan, our construction represents the Sullivan completions as homotopy limits of idempotent functors. In addition, we show that the Sullivan completion is idempotent on those spaces (in analogy with the Bousfield and Kan completion on good spaces) for which its cohomology with coefficients in agrees with that of our ArtinMazur style completion. Finally, we rigidify the ArtinMazur completion to obtain an idempotent ArtinMazur completion on a category of generalized prospaces which preserves fibration and suitably defined cofibration sequences. (Our previous results on idempotency and factorization lift to the rigid completion.) Our results answer questions of Adams, Sullivan, and, later, A. Deleanu.
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 J. F. Adams, Adams's problems, ManifoldsTokyo 1973, Univ. of Tokyo Press, Tokyo, 1975, pp. 430431. MR 0375289 (51:11485)
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 J. L. Aronson and D. A. Edwards, The mathematical foundations of quantum mechanics, Synthèse 42 (1979), 170. MR 549604 (80j:81006)
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 M. Artin and B. Mazur, Étale homotopy theory, Lecture Notes in Math., Vol. 100, Springer, New York, 1969. MR 0245577 (39:6883)
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 A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133150. MR 0380779 (52:1676)
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 A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Math., Vol. 304, Springer, New York, 1973. MR 0365573 (51:1825)
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 R. Brown and P. R. Heath, Coglueing homotopy equivalences, Math. Z. 113 (1970), 313325. MR 0266213 (42:1120)
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 A. Deleanu, Topologized objects in categories and the Sullivan profinite completion, J. Pure Appl. Algebra 25 (1982), 2124. MR 660387 (83i:55012)
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 A. Deleanu and P. Hilton, Generalized shape theory, General Topology and Its Relations to Modern Analysis and Algebra. IV (J. Novak, ed.), Lecture Notes in Math., Vol. 609, Springer, New York, 1977, pp. 5665. MR 0506660 (58:22247)
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 D. A. Edwards and R. Geoghegan, The stability problem in shape and a Whitehead theorem in prohomotopy, Trans. Amer. Math. Soc. 214 (1975), 261277. MR 0413095 (54:1216)
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 D. A. Edwards and H. M. Hastings, Čech and Steenrod homotopy theory with applications to geometric topology, Lecture Notes in Math., Vol. 542, Springer, New York, 1976. MR 0428322 (55:1347)
 [13]
 , Why the completion works, General Topology Appl. 7 (1977), 179184. MR 0454966 (56:13208)
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 [15]
 , Steenrod homotopy theory, homotopy idempotents, and homotopy limits, Topology Proc. 2 (1977), 461477. MR 540623 (80k:55035)
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 Ju. T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape, Zagreb, 1982, preprint.
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 S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math., vol. 5, Springer, New York, 1971. MR 1712872 (2001j:18001)
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 S. Mardešić, On the Whitehead theorem in shape theory. I, II, Fund. Math. 91 (1976), 5164, 93103. MR 0407798 (53:11568)
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 M. Mosynska, The Whitehead theorem in the theory of shapes, Fund. Math. 80 (1973), 235240. MR 0339159 (49:3922)
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DOI:
http://dx.doi.org/10.1090/S00029947198507662269
PII:
S 00029947(1985)07662269
Article copyright:
© Copyright 1985
American Mathematical Society
