Cotriple homology

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 formula_1. Let "F" be the left adjoint of the forgetful functor from the category of rings to Set; i.e., free module functor. Then formula_2 defines a cotriple and the "n"-th cotriple homology of formula_3 is the "n"-th left derived functor of "E" evaluated at "M"; i.e., formula_4.
Example (algebraic K-theory): Let us write "GL" for the functor formula_5. As before, formula_2 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.