# Dimension quotients, Fox subgroups and limits of functors

## TL;DR

This paper provides a functorial framework to describe the fourth dimension quotient and third Fox subgroup of a group, linking these to limits of functors and derived quadratic functors.

## Contribution

It introduces a functorial approach to describe dimension quotients and Fox subgroups, offering new identifications and representations without isolators.

## Key findings

- Describes the fourth dimension quotient via limits of functors.
- Provides a functorial description of a quotient of the third Fox subgroup.
- Shows the limit over free representations relates to derived quadratic functors.

## Abstract

This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of the third Fox subgroup is given and, as a consequence, an identification (not involving an isolator) of the third Fox subgroup is obtained. It is shown that the limit over the category of free representations of the third Fox quotient represents the composite of two derived quadratic functors.