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

TL;DR

This paper constructs an embedding of the category of graphs into abelian groups, showing the latter's complexity and using it to address longstanding problems related to subcategories and large cardinal axioms.

Contribution

It introduces a functorial embedding of graphs into abelian groups that reveals the category's complexity and applies it to solve classical categorical problems.

Findings

The embedding demonstrates the category of abelian groups is as complex as any concrete category.

It provides a positive solution to Isbell's problem under large cardinal assumptions.

Several known constructions in abelian groups are derived straightforwardly from the embedding.

Abstract

We construct an embedding G of the category of graphs into the category of abelian groups such that for graphs X and Y we have Hom(GX,GY)=Z[Hom(X,Y)], the free abelian group whose basis is the set Hom(X,Y). The isomorphism is functorial in X and Y. The existence of such an embedding implies that, contrary to a common belief, the category of abelian groups is as complex and comprehensive as any other concrete category. We use this embedding to settle an old problem of Isbell whether every full subcategory of the category of abelian groups, which is closed under limits, is reflective. A positive answer turns out to be equivalent to weak Vopenka's principle, a large cardinal axiom which is not provable but believed to be consistent with standard set theory. Several known constructions in the category of abelian groups are obtained as quick applications of the embedding. In the revised version we add some consequences to the Hovey-Palmieri-Stricland problem about existence of arbitrary localizations in a stable homotopy category