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

Abstract

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.

The Complexity of the Embeddability Relation Between Torsion-Free Abelian Groups of Uncountable Size

“…We recall one more theorem of existence of R-algebras with prescribed endomorphism ring, that is a result of the same kind of Theorem 3.4. Theorem 3.6 (Göbel-Przeździecki [10,Corollary 4.5]). Let R be an S-cotorsion-free ring such that |R| < 2 ℵ 0 .…”

“…For any ϕ i and S ∈ [T ] <ω 1 , let δ S i be the function such that the diagram commutes Remark 4.7. Theorem 4.6 states that G is an almost-full embedding, according to the terminology of [10,18]. It can be proved arguing as in [18,Section 3].…”

“…Section 4 is dedicated to the proof of Theorem 1.2: we define a Borel reduction from the embeddability relation on κ-sized graphs to embeddability on κ-sized torsion-free abelian groups. In Section 5 we follow the main ideas of [GP14], and we exploit the techniques used in Section 4 to prove an analogue of Theorem 1.2 for the embeddability relation on R-modules, for every S-cotorsion-free ring R of cardinality less than the continuum. Theorem 1.3.…”

“…Then one can argue as in Lemmas 4.4, 4.5, and 4.8 to prove that G is a Borel reduction from κ GRAPHS to κ R-MOD . In this case the almost-fullness for G was proved essentially in [10,Theorem 3.16] with the same argument used in [18]. Corollary 5.1.…”