An axiomatic construction of an almost full embedding of the category of graphs into the category of \(R\)-objects (Q392398)

The authors construct embeddings \(G\) to the category of graphs into categories of \(R\)-modules for all commutative, cotorsion-free rings \(R.\) These embeddings are almost full in the sense that \(R[\Hom_{\text{Graphs}}(X,Y)] \rightarrow \Hom_R(GX,GY)\) are isomorphisms, where \(R[S]\) denotes the free \(R\)-module with the basis \(S\). In other words, every morphism in the target is an \(R\)-linear combination of morphisms coming from the source. In particular the authors obtain an almost full embedding of the category of graphs into the category of vector spaces with four distinguished subspaces (over any field).