Relation Morphisms of Directed Graphs

Abstract Associating graph algebras to directed graphs leads to both covariant and contravariant functors from suitable categories of graphs to the category $k$-$\textsf{Alg}$ of algebras and algebra homomorphisms. As both functors are often used at the same time, finding a new category of graphs that allows a “common denominator” functor unifying the covariant and contravariant constructions is a fundamental problem. Herein, we solve this problem by first introducing the relation category of graphs $\textsf{RG}$, and then determining the concept of admissible graph relations that yields a subcategory of $\textsf{RG}$ admitting a contravariant functor to $k$-$\textsf{Alg}$ simultaneously generalizing the aforementioned covariant and contravariant functors. Although we focus on Leavitt path algebras and graph C*-algebras, on the way we unravel functors to $k$-$\textsf{Alg}$ given by path algebras, Cohn path algebras, and Toeplitz graph C*-algebras from suitable subcategories of $\textsf{RG}$. Better still, we illustrate relation morphisms of graphs by naturally occurring examples, including Cuntz algebras, quantum spheres, and quantum balls.