Realizing diagrams in the homotopy category by means of diagrams of simplicial sets
Authors:
W. G. Dwyer and D. M. Kan
Journal:
Proc. Amer. Math. Soc. 91 (1984), 456460
MSC:
Primary 55P15; Secondary 55U35
MathSciNet review:
744648
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Abstract: Given a small category , we show that a diagram in the homotopy category can be realized by a diagram of simplicial sets iff a certain simplicial set is nonempty. Moreover, this simplicial set can be expressed as the homotopy inverse limit of simplicial sets whose homtopy types are quite well understood. There is also an associated obstruction theory. In the special case that is a group (i.e. has only one object and all its maps are invertible) these results reduce to the ones of G. Cooke.
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 [2]
 G. Cooke, Replacing homotopy actions by topological actions, Trans. Amer. Math. Soc. 237 (1978), 391406. MR 0461544 (57:1529)
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 , A classification theorem for diagrams of simplicial sets, Topology (to appear). MR 744846 (86c:55010a)
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 , An obstruction theory for diagrams of simplicial sets, Nederl. Akad. Wetenesch. Proc. Ser. A 87 = Indag. Math. 46 (1984). MR 749527 (86c:55010c)
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 J. P. May, Simplicial objects in algebraic topology, Van Nostrand, Princeton, N.J., 1967. MR 0222892 (36:5942)
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 D. G. Quillen, Higher algebraic theory. I, Lecture Notes in Math., vol. 341, SpringerVerlag, Berlin and New York, 1973, pp. 85147. MR 0338129 (49:2895)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407446484
PII:
S 00029939(1984)07446484
Article copyright:
© Copyright 1984
American Mathematical Society
