Higher limits, homology theories and fr-codes

TL;DR

This paper introduces a framework for understanding limits over presentations, explores homology theories via derived limits, and develops fr-codes as a symbolic method to encode various functors from groups to abelian groups.

Contribution

It presents a novel approach to coding functors using fr-codes and extends the theory of limits in the context of homology and derived functors.

Findings

Development of the theory of limits over presentations

Introduction of fr-codes for functor encoding

Application to homology and derived functors

Abstract

This text is based on lectures given by authors in summer 2015. It contains an introduction to the theory of limits over the category of presentations, with examples of different well-known functors like homology or derived functors of non-additive functors in a form of derived limits. The theory of so-called ${\bf fr}$-codes also is developed. This is a method how different functors from the category of groups to the category of abelian groups, such as group homology, tensor products of abelianization, can be coded as sentences in the alphabet with two symbols ${\bf f}$ and ${\bf r}$.