An “almost” full embedding of the category of graphs into the category of groups

Abstract

We construct a functor F : G raphs → G roups 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 FY and a homomorphism of the form Ff, 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.