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.
 [1]
A.
K. Bousfield and D.
M. Kan, Homotopy limits, completions and localizations,
Lecture Notes in Mathematics, Vol. 304, SpringerVerlag, Berlin, 1972. MR 0365573
(51 #1825)
 [2]
George
Cooke, Replacing homotopy actions by
topological actions, Trans. Amer. Math.
Soc. 237 (1978),
391–406. MR 0461544
(57 #1529), http://dx.doi.org/10.1090/S00029947197804615442
 [3]
W.
G. Dwyer and D.
M. Kan, Function complexes in homotopical algebra, Topology
19 (1980), no. 4, 427–440. MR 584566
(81m:55018), http://dx.doi.org/10.1016/00409383(80)900257
 [4]
W.
G. Dwyer and D.
M. Kan, Function complexes for diagrams of simplicial sets,
Nederl. Akad. Wetensch. Indag. Math. 45 (1983),
no. 2, 139–147. MR 705421
(85e:55038)
 [5]
W.
G. Dwyer and D.
M. Kan, A classification theorem for diagrams of simplicial
sets, Topology 23 (1984), no. 2, 139–155.
MR 744846
(86c:55010a), http://dx.doi.org/10.1016/00409383(84)900351
 [6]
W.
G. Dwyer and D.
M. Kan, Equivariant homotopy classification, J. Pure Appl.
Algebra 35 (1985), no. 3, 269–285. MR 777259
(86h:55008), http://dx.doi.org/10.1016/00224049(85)900453
 [7]
W.
G. Dwyer and D.
M. Kan, An obstruction theory for diagrams of simplicial sets,
Nederl. Akad. Wetensch. Indag. Math. 46 (1984),
no. 2, 139–146. MR 749527
(86c:55010c)
 [8]
Daniel
M. Kan, On c. s. s. complexes, Amer. J. Math.
79 (1957), 449–476. MR 0090047
(19,759e)
 [9]
J.
Peter May, Simplicial objects in algebraic topology, Van
Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc.,
Princeton, N.J.Toronto, Ont.London, 1967. MR 0222892
(36 #5942)
 [10]
Daniel
Quillen, Higher algebraic 𝐾theory. I, Algebraic
𝐾theory, I: Higher 𝐾theories (Proc. Conf., Battelle
Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973,
pp. 85–147. Lecture Notes in Math., Vol. 341. MR 0338129
(49 #2895)
 [1]
 A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, SpringerVerlag, Berlin and New York, 1972. MR 0365573 (51:1825)
 [2]
 G. Cooke, Replacing homotopy actions by topological actions, Trans. Amer. Math. Soc. 237 (1978), 391406. MR 0461544 (57:1529)
 [3]
 W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427440. 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), 449476. 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 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
