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

TL;DR

This paper constructs an almost full embedding of the category of graphs into categories of R-modules, showing that R-module categories are as complex as graph categories for cotorsion-free rings.

Contribution

It introduces an axiomatic method to embed graphs into R-module categories with near full faithfulness, revealing their comparable complexity.

Findings

Embedding is almost full: induced maps are isomorphisms.

Categories of R-modules are as complex as graph categories for cotorsion-free rings.

Similar embedding into vector spaces with four subspaces over any field.

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).