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

Example (algebraic K-theory):[1] Let us write GL for the functor ${\displaystyle R\mapsto \varinjlim _{n}GL_{n}(R)}$. As before, ${\displaystyle FU}$ defines a cotriple on the category of rings with F free ring functor and U forgetful. For a ring R, one has:

${\displaystyle K_{n}(R)=\pi _{n-2}GL(FU_{*}R),\,n\geq 3}$ 

where on the left is the n-th K-group of R. This example is an instance of nonabelian homological algebra.

Notes

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.

References

• 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.

Further reading

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