Cotriple homology explained

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

E =- ⊗ _{R} N . Let F be the left adjoint of the forgetful functor from the category of rings to Set; i.e., free module functor. Then FU defines a cotriple and the n-th cotriple homology of E(FU _{*M)} is the n-th left derived functor of E evaluated at M; i.e., \operatorname{Tor} ^{R} _{n(M,} N) .

Example (algebraic K-theory):^[1] Let us write GL for the functor

R \mapsto \varinjlim _{n} GL _{n(R)} . As before, FU defines a cotriple on the category of rings with F free ring functor and U forgetful. For a ring R, one has:K _{n(R)} = \pi _{n-2} GL(FU _{*} R), n \ge 3  where on the left is the n-th K-group of R. This example is an instance of nonabelian homological algebra.

Further reading

• Who Threw a Free Algebra in My Free Algebra?, a blog post.

Notes and References

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