Cotriple homology

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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):[1] 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[edit]

  1. ^ Swan, Richard G. (1972). "Some relations between higher K-functors". Journal of Algebra. 21: 113–136. doi:10.1016/0021-8693(72)90039-7.

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