The Right Angled Artin Group Functor as a Categorical Embedding

It has long been known that the combinatorial properties of a graph $\mathrm{\Gamma }$ are closely related to the group theoretic properties of its right angled artin group (raag). It’s natural to ask if the graph homomorphisms are similarly related to the group homomorphisms between two raags. The main result of this paper shows that there is a purely algebraic way to characterize the raags amongst groups, and the graph homomorphisms amongst the group homomorphisms. As a corollary we present a new algorithm for recovering $\mathrm{\Gamma }$ from its raag.