An "almost" full embedding of the category of graphs into the category of groups

TL;DR

This paper constructs a functor from graphs to groups that is nearly full and faithful, enabling new applications in localizations and homotopy theory within group categories.

Contribution

It introduces an 'almost' full and faithful functor from graphs to groups, bridging the two categories with novel applications in localizations and homotopy theory.

Findings

The functor is faithful and 'almost' full, inducing bijections up to trivial homomorphisms and conjugation.

Applications include new insights into localizations in group and homotopy categories.

Provides a framework for translating graph properties into group-theoretic contexts.

Abstract

We construct a functor from the category of graphs to the category of groups which is faithful and "almost" full, in the sense that it induces bijections of the Hom sets up to trivial homomorphisms and conjugation in the category of groups. We provide several applications of this construction to localizations (i.e. idempotent functors) in the category of groups and the homotopy category.