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

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, Springer-Verlag, Berlin-New York, 1972. MR**0365573****[2]**George Cooke,*Replacing homotopy actions by topological actions*, Trans. Amer. Math. Soc.**237**(1978), 391–406. MR**0461544**, 10.1090/S0002-9947-1978-0461544-2**[3]**W. G. Dwyer and D. M. Kan,*Function complexes in homotopical algebra*, Topology**19**(1980), no. 4, 427–440. MR**584566**, 10.1016/0040-9383(80)90025-7**[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****[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**, 10.1016/0040-9383(84)90035-1**[6]**W. G. Dwyer and D. M. Kan,*Equivariant homotopy classification*, J. Pure Appl. Algebra**35**(1985), no. 3, 269–285. MR**777259**, 10.1016/0022-4049(85)90045-3**[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****[8]**Daniel M. Kan,*On c. s. s. complexes*, Amer. J. Math.**79**(1957), 449–476. MR**0090047****[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****[10]**Daniel Quillen,*Higher algebraic 𝐾-theory. I*, Algebraic 𝐾-theory, I: Higher 𝐾-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR**0338129**

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0744648-4

Article copyright:
© Copyright 1984
American Mathematical Society