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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), 456-460
MSC: Primary 55P15; Secondary 55U35
DOI: https://doi.org/10.1090/S0002-9939-1984-0744648-4
MathSciNet review: 744648
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Abstract: Given a small category $ {\mathbf{D}}$, we show that a $ {\mathbf{D}}$-diagram $ \bar X$ in the homotopy category can be realized by a $ {\mathbf{D}}$-diagram of simplicial sets iff a certain simplicial set $ r\bar X$ is nonempty. Moreover, this simplicial set $ r\bar X$ 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 $ {\mathbf{D}}$ is a group (i.e. $ {\mathbf{D}}$ has only one object and all its maps are invertible) these results reduce to the ones of G. Cooke.


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  • [1] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin and New York, 1972. MR 0365573 (51:1825)
  • [2] G. Cooke, Replacing homotopy actions by topological actions, Trans. Amer. Math. Soc. 237 (1978), 391-406. MR 0461544 (57:1529)
  • [3] W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440. MR 584566 (81m:55018)
  • [4] -, Function complexes for diagrams of simplicial sets, Nederl. Akad. Wetensch. Proc. Ser. A 86 = Indag. Math. 45 (1983). MR 705421 (85e:55038)
  • [5] -, A classification theorem for diagrams of simplicial sets, Topology (to appear). MR 744846 (86c:55010a)
  • [6] -, Equivariant homotopy classification, J. Pure Appl. Algebra (to appear). MR 777259 (86h:55008)
  • [7] -, An obstruction theory for diagrams of simplicial sets, Nederl. Akad. Wetenesch. Proc. Ser. A 87 = Indag. Math. 46 (1984). MR 749527 (86c:55010c)
  • [8] D. M. Kan, On c.s.s. complexes, Amer. J. Math. 79 (1957), 449-476. MR 0090047 (19:759e)
  • [9] J. P. May, Simplicial objects in algebraic topology, Van Nostrand, Princeton, N.J., 1967. MR 0222892 (36:5942)
  • [10] D. G. Quillen, Higher algebraic $ K$-theory. I, Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin and New York, 1973, pp. 85-147. MR 0338129 (49:2895)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0744648-4
Article copyright: © Copyright 1984 American Mathematical Society

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