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In mathematics, a homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. The homotopy category of all topological spaces is often denoted hTop or Toph. ==Definition and examples== The homotopy category hTop of topological spaces is the category whose objects are topological spaces. Instead of taking continuous functions as morphisms between two such spaces, the morphisms in hTop between two spaces ''X'' and ''Y'' are given by the equivalence classes of all continuous functions ''X'' → ''Y'' with respect to the relation of homotopy. That is to say, two continuous functions are considered the same morphism in hTop if they can be deformed into one another via a (continuous) homotopy. The set of morphisms between spaces ''X'' and ''Y'' in a homotopy category is commonly denoted () rather than Hom(''X'',''Y''). The composition :(''Y'' ) × (''Z'' ) → (''Z'' ) is defined by :() o () = (o ''g'' ). This is well-defined since the homotopy relation is compatible with function composition. That is, if ''f''1, ''g''1 : ''X'' → ''Y'' are homotopic and ''f''2, ''g''2 : ''Y'' → ''Z'' are homotopic then their compositions ''f''2 o ''f''1, ''g''2 o ''g''1 ''X'' → ''Z'' are homotopic as well. While the objects of a homotopy category are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. Indeed, hTop is an example of a category that is not concretizable, meaning there does not exist a faithful forgetful functor :''U'' : hTop → Set to the category of sets. Homotopy categories are examples of quotient categories. The category hTop is a quotient of Top, the ordinary category of topological spaces. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homotopy category」の詳細全文を読む スポンサード リンク
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