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

TL;DR

This paper demonstrates that the category of simple graphs can be fully embedded and is coreflective within a specific comma category involving groups and free group functors, generalizing a known embedding technique.

Contribution

It introduces a general method for embedding categories into comma categories and shows simple graphs are coreflective in the comma category of groups under the free group functor.

Findings

Simple graphs form a full coreflective subcategory

Embedding techniques extend to topological spaces and other categories

Provides a new perspective on categorical relationships involving graphs and groups

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 $(\mathcal{F}\downarrow\mathbf{Grp})$.