Topic summary
Homotopy category of topological spaces
Extracted from the Wikipedia article Homotopy category.
Model categories
The same homotopy category can arise from many different model categories. An important example is the standard model structure on simplicial sets: the associated homotopy category is equivalent to the homotopy category of topological spaces, even though simplicial sets are combinatorially defined objects that lack any topology. Some topologists prefer instead to work with compactly generatedweak Hausdorff spaces; again, with the standard model structure, the associated homotopy category is equivalent to the homotopy category of all topological spaces.