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., .
where on the left is the n-th K-group of R. This example is an instance of nonabelian homological algebra.
- Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
- Who Threw a Free Algebra in My Free Algebra?, a blog post.
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