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.
Let F be the left adjoint of the forgetful functor from the category of rings to Set; i.e., free module functor.
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
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.
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