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 havethe free abelian group whose basis is the set Hom Graphs (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 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-Stricland 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.MSC: 18B15 (20K20,20K30,18A40,03E55)
We construct a functor F : Graphs → Groups which is faithful and "almost" full, in the sense that every nontrivial group homomorphism F X → F Y is a composition of an inner automorphism of F Y and a homomorphism of the form F f , for a unique map of graphs f : X → Y . When F is composed with the Eilenberg-Mac Lane space construction K(F X, 1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:(1) Is every orthogonality class reflective?(2) Is every orthogonality class a small-orthogonality class? have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopěnka's principle and (2) to Vopěnka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopěnka's principle.MSC: 18A40; 20J15; 55P60; 18A22; 03E55
An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduce the problem to infinite metrizable profinite groups. We provide a positive answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.2010 Mathematics Subject Classification. 28C10, 28A05, 22C05, 03E17.
We construct embeddings G of the category of graphs into categories of Rmodules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of Gare isomorphisms. The symbol R[S] above denotes the free R-module with the basis S. This implies that, for any cotorsion-free ring R, the categories of Rmodules are not less complicated than the category of graphs. A similar embedding of graphs into the category of vector spaces with four distinguished subspaces (over any field, e.g. F 2 = {0, 1}) is obtained.
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