Dimension quotients as boundary limits

TL;DR

This paper introduces the concept of boundary limits for functors from free presentations of groups and demonstrates that the fourth dimension quotient of a group can be characterized as such a boundary limit.

Contribution

It provides a new functorial description of the fourth dimension quotient using boundary limits, linking group theory and categorical limits.

Findings

Boundary limit concept applied to group functors

Fourth dimension quotient characterized as a boundary limit

Provides a new categorical perspective on group invariants

Abstract

For a functor from the category of free presentations of a group to the category of all groups we define the boundary limit as an image of the natural map from limit to colimit. We show that the fourth dimension quotient of a group can be naturally described as the boundary limit of a simply-defined functor.