The category of simple graphs is coreflective in the comma category of groups under the free group functor

Abstract

We show that the comma category $(\mathcal{F}\downarrow\mathbf{Grp})$ of groups under the free group functor $\mathcal{F}: \mathbf{Set} \to \mathbf{Grp}$ contains the category $\mathbf{Gph}$ of simple graphs as a full coreflective subcategory. More broadly, we generalize the embedding of topological spaces into Steven Vickers' category of topological systems to a simple technique for embedding certain categories into comma categories, then show as a straightforward application that simple graphs are coreflective in $(\mathcal{F}\downarrow\mathbf{Grp})$.