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

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 categoryComment: 20 page

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

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.Comment: 24 pages; to appear in Adv. Math