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In algebra, given a category ''C'' with a cotriple, the ''n''-th cotriple homology of an object ''X'' in ''C'' with coefficients in a functor ''E'' is the ''n''-th homotopy group of the ''E'' of the augmented simplicial object induced from ''X'' by the cotriple. The term "homology" is because in the abelian case, by the Dold–Kan correspondence, the homotopy groups are the homology of the corresponding chain complex. Example: Let ''N'' be a left module over a ring ''R'' and let . Let ''F'' be the left adjoint of the forgetful functor from the category of rings to Set; i.e., free module functor. Then defines a cotriple and the ''n''-th cotriple homology of is the ''n''-th left derived functor of ''E'' evaluated at ''M''; i.e., . Example (algebraic K-theory):〔Swan, (Some relations between K-functors )〕 Let us write ''GL'' for the functor . As before, defines a cotriple on the category of rings with ''F'' free ring functor and ''U'' forgetful. For a ring ''R'', one has: : where on the left is the ''n''-th ''K''-group of ''R''. This example is an instance of nonabelian homological algebra. == Notes == 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「Cotriple homology」の詳細全文を読む スポンサード リンク
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