An axiomatic construction of an almost full embedding of the category of graphs into the category of R-objects

Abstract

We construct embeddings G of the category of graphs into categories of R-modules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of G R[Hom_Graphs(X,Y)] --> Hom_R(GX,GY) are 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 R-modules 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).