nLab fr-code

Idea

In homological algebra it turns out that a host of common (co)homological constructions (such as group homology, cyclic homology, etc.) may be cast in a unified way as homotopy limits of functors on categories of presentations of the given algebraic structure (group, Lie algebra, associative algebra, etc.) [Ivanov & Mikhailov 2015], in fact all these functors may systematically be indexed by “fr-codes” [Ivanov & Mikhailov 2017].

Construction

Consider $\mathbb{Z}F\colon Pres\to Ring$, a functor of rings on the category of all free extensions of the form $1\to R\to F\to G\to 1$, which takes a free extension (free presentations) and sends it to the group ring $\mathbb{Z}F$. There are two functorial ideals $\mathbf{f}$ and $\mathbf{r}$ in the (functorial) ring $\mathbb{Z}F$ that are defined as follows:

$$\mathbf{f} \;\coloneqq\; ker(\mathbb{Z}F \to \mathbb{Z}) \,,$$

$$\mathbf{r} \;\coloneqq\; ker(\mathbb{Z}F \to \mathbb{Z}G) \,.$$

That is, $\mathbf{f}$ is the augmentation ideal of the group $F$, and it is generated by expressions of the form $(w - 1)$ where $w \in F$, and $\mathbf{r}$ is a sub-ideal of $\mathbf{f}$ generated by expressions of the form $(r - 1)$ where $r \in R$.

Since $\mathbf{f},\mathbf{r}$ are ideals of the functor of rings $ZF$, one may form sums and intersections of monomials: $\mathbf{rf}\cap \mathbf{fr}, \mathbf{r+ff}, \mathbf{r}^{k+1}+\mathbf{f}\mathbf{r}^k\mathbf{f}, \dots$

These are functors on the category of free extensions $Pres$ with values in abelian groups.

Let $Pres(G)\subset Pres$ denote a fiber of a functor $Pres\to Gr$ sending a free extension $1\to R\to F\to G\to 1$ to $G$. Given an $\mathbf{f\r}$-expression $w(\mathbf{f},\mathbf{r})$ and a group $G$, one can define

$${}^i[w(\mathbf{f},\mathbf{r})](G) := lim^i (w(\mathbf{f},\mathbf{r})|_{Pres(G)}).$$

where $lim^i (\mathcal{F}: C\to Ab)$ denotes the $i$-th right derived functor of the limit functor $lim: Fun(C,Ab)\to Ab$.

It turns out that by exploiting some features of the category $Pres(G)$ this construction can be made functorial in group $G$. The first such feature is that it has all binary coproducts (in particular, its classifying space is contractible). That feature is used in by authors of [Ivanov & Mikhailov 2015], [Ivanov & Mikhailov 2017] extensively, since it ensures triviality of higher limits of constant functors.

Secondly, this category is strongly connected, in that the hom-set ${\sf hom}(c,c')$ is not empty for any pair of objects $c$ and $c'$. Hence, with each $\mathbf{f\r}$-expression $w(\mathbf{f},\mathbf{r})$ we associate a graded functor from the category Grp of groups to the category Ab of abelian groups, ${}^i[w(\mathbf{f},\mathbf{r})] \colon Gr \to Ab\,.$

In [Golub 2024] author suggests a homotopy theoretic construction not using the homological algebra.

Related entries

• higher limit approach to homology

References

The original articles

• 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]

• Sergei O. Ivanov, Roman Mikhailov: Higher limits, homology theories and $\mathbf{fr}$-codes, in: Combinatorial and Toric Homotopy, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (2017) 229-261 [arXiv:1510.09044, doi:10.1142/9789813226579_0004]

• Sergei O. Ivanov, Roman Mikhailov, Fedor Pavutnitskiy: Limits, standard complexes and $\mathbf{fr}$-codes, Sb. Math. 211 (2020) 1568 [arXiv:1906.08793, doi:10.1070/SM9348, doi:10.1070/SM9348]

Further discussion:

• Nikita Golub: Functorial languages in homological algebra and the lower central series [arXiv:2410.05708]

• Nikita Golub: Functorial Languages in Homological Algebra, talk at CQTS @ NYU Abu Dhabi (Oct 2024) [slides:pdf]