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].

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]