Cotriple homology

Short description: Concept in category theory

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

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

[math]\displaystyle{ K_n(R) = \pi_{n-2} GL(FU_* R), \, n \ge 3 }[/math] 

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

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. OCLC 36131259. 

Further reading

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