|
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category. ==Introduction and motivation== A category ''C'' consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace ''C'' by another category ''C in which certain morphisms are forced to be isomorphisms. This process is called localization. For example, in the category of ''R''-modules (for some fixed commutative ring ''R'') the multiplication by a fixed element ''r'' of ''R'' is typically (i.e., unless ''r'' is a unit) not an isomorphism: : The category that is most closely related to ''R''-modules, but where this map ''is'' an isomorphism turns out to be the category of -modules. Here is the localization of ''R'' with respect to the (multiplicatively closed) subset ''S'' consisting of all powers of ''r'', The expression "most closely related" is formalized by two conditions: first, there is a functor : sending any ''R''-module to its localization with respect to ''S''. Moreover, given any category ''C'' and any functor : sending the multiplication map by ''r'' on any ''R''-module (see above) to an isomorphism of ''C'', there is a unique functor : such that . 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「localization of a category」の詳細全文を読む スポンサード リンク
|