nLab A higher limit approach

Idea

In homological algebra one usually studies functors of the form $\mathcal{A}\to Ab, k-mod, \dots$ defined via derived functors where $\mathcal{A}$ is some category of groups, algebras over a ring $k$ etc., and $Ab$, $k-Mod$ are categories of abelian groups and $k$-modules, where $k$ is some ground ring.

For instance, for the category of groups $\mathcal{A}=$ Grp, using the left derived functors $L_q(\mathbb{Z}\otimes_{\mathbb{Z}G}(-))$ of the functor of coinvariants $\mathbb{Z}\otimes_{\mathbb{Z}G}(-): \mathbb{Z}G-mod\to Ab$ one usually defines group homology $H_q(G;A) =L_q(\mathbb{Z}\otimes_{\mathbb{Z}G}(-))(A)$ of a group $G$ with coefficients in a $\mathbb{Z}G$-modules $A$.

The idea of a higher limit approach is to use categories $Pres(A)$ of extensions $0\to I\to P\to A\to 0$ where $P$ is a projective object in a category of algebraic objects $\mathcal{A}$ (e.g free presentations of groups) and describe such functors of homological nature $H:\mathcal{A}\to Ab, k-mod,\dots$ using derived/higher (co)limits of some simple functors $\mathcal{F}$ from the category $H\simeq lim^{*}, colim_{*}(\mathcal{F} \colon Pres(A)\to Ab, k-mod, \dots)$.

This approach originates in the work [Quillen 1989], where, for instance, he derived the formula:

$$HC_{2n}(A) \;=\; lim\big( F/(I^{n+1}+[F,F]) \big) \,,$$

where $HC_{2n}(A)$ is cyclic homology of $A$ over a field of characteristic zero $k$ and $F/(I^{n+1}+[F,F])$ takes a free algebra extension $0\to I\to F\to A\to 0$ and sends it to the abelian group $F/(I^{n+1}+[F,F])$.

References

The original articles

• Daniel Quillen: Cyclic cohomology and algebra extensions, K-Theory v. 3, n. 3 (1989): 205-246 [doi:10.1007/BF00533370]

• Roman Mikhailov, Ioannis Emmanouil: A limit approach to group homology, Journal of Algebra

  Volume 319, Issue 4, 15 February 2008, Pages 1450-1461 [doi:10.1016/j.jalgebra.2007.12.006]

• Sergei O. Ivanov, Roman Mikhailov: A higher limit approach to homology theories, Journal of Pure and Applied Algebra 219 6 (2015) 1915-1939 [arXiv:1309.4920, doi:10.1016/j.jpaa.2014.07.016]