Smash – Hom Duality for Pointed Simple Reflexive Graphs

Abstract

Graph theory has a wide range of applications in natural sciences such as biology, chemistry, physics, computer science, engineering, etc. Since the second half of the last century, mathematicians have tended to rewrite existing mathematics in the language of category theory, as it provides a more transparent and conceptual foundation. Other advantages of this approach, over set theory, include uncovering previously unseen relationships and providing a gateway to transport ideas between different fields of mathematics. Following this idea, we apply some categorical concepts to graph theory in this research. First, we recognize a new product for pointed simple reflexive graphs, and the binary product in the category of such graphs is called the smash product. We also realize the set of based point preserving graph homomorphisms between two given pointed simple reflexive graphs as another such graph. Finally, we establish a duality result between smash product and graph of based graph homomorphism, which is reminiscent of the classical smash-hom adjunction for pointed topological spaces. In future work, we will show this correspondence is a genuine adjunction and prove that simple reflexive graphs form a symmetric monoidal category with the smash product.