By David A. Edwards, Harold M. Hastings (auth.)
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Additional resources for Čech and Steenrod Homotopy Theories with Applications to Geometric Topology
Ay-l~)ef. 3]. X PX(n) It is easy to check that the definitions are equivalent. 43) o and o' Definition. of X Sp. Let X be a Ken simplicial spectrum. , are homotopic, denoted [May - i, Def. 1]). 44) for Proposition. spectrum, then i ~ 2. 2]). If is an equivalence relation on the k - simplices of X X, is a Ken for all k. Proof. 45) spectrum. 2]. Definition. Let ~_ (Compare [May-l, Def. 6]). Let denote the set of all of the k - s i m p l i c e s X be a Kan a of X which 32 satisfy dlo = * for all i.
Quillen [Q- i, w N(1) For the usual product. is trivial. N(3), let N(2) is 46 Top; the category of topological spaces with the following structure: cofibrations and fibrations are defined by the homotopy-extension and covering-homotopy properties, respectively; weak equivalences are ordinary homotopy equivalences. Condition CG; N This is due to A. Str6m [Str]. is clear. the category of compactly generated spaces, with a similar structure. See N. E. Steenrod [St -3]; also, Sing; [Has -3]. the category of topological spaces with the following singular structure: of CW cofibrations are pushouts of inclusions of w complexes, fibrations are Serre fibrations, weak equivalences are weak homotopy equivalences [Q -i, w Again, Condition N is clear.
5] are easily veri- Their precise statement and proof is omitted. 32) Remarks. Adams defines an internal mapping functor ing to Brown's Theorem (see [Adams - i]). HOM by appeal- This approach also yields the above theorem. This concludes our formulation of the category of CW spectra. We shall now briefly discuss simpllclal spectra and the equivalence of homotopy categories. the end of this section we shall use CW At spectra to discuss homology and cohomology theories and operations, following [Adams -1,3].
Čech and Steenrod Homotopy Theories with Applications to Geometric Topology by David A. Edwards, Harold M. Hastings (auth.)