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Field: Topology |
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- First of all, this article has no references. It also presents as rather stream-of-consciousness, with neither an introductory overview nor a conceptual flow. The context given is mainly of interest to specialists, and the information comes off as having been chosen haphazardly. It could be either a weak "Start" or a strong "Stub", but I think it needs a lot of reorganization, expansion, and polishing. Ryan Reich 01:35, 6 July 2007 (UTC)
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[edit] Homotopy categories of model categories.
It seems to me that this description of a homotopy category is too narrow. From the current perspective homotopy categories are usually understood in the context of model categories. For a model category M its homotopy category is equivalent to the category constructed by taking the full subcategory of M on fibrant-cofibrant objects and then inverting all weak equivalences. As a result one does not need to argue which homotopy category is "the" homotopy category - there is one for each model category. One ("Serre") model category structure on the category of topological spaces yields the homotopy category equivalent to the category of CW-complexes and homotopy classes of maps between CW-complexes. Another ("Hurewicz") model category structure on topological spaces produces the homotopy category equivalent to the category of all spaces with homotopy classes of maps as morphisms. --69.204.54.113 (talk) 05:27, 22 August 2008 (UTC)
