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

Abstract

We construct an embedding G:Graphs→Ab of the category of graphs into the category of abelian groups such that for X and Y in Graphs we haveHom(GX,GY)≅Z[HomGraphs(X,Y)], the free abelian group whose basis is the set HomGraphs(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 as of 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 Vopěnka's principle, a large cardinal axiom which is not provable but believed to be consistent with standard set theory. We obtain some consequences to the Hovey–Palmieri–Strickland problem about existence of arbitrary localizations in a stable homotopy category.Several known constructions in the category of abelian groups are obtained as quick applications of the embedding.