On the Concrete Categories of Graphs

TL;DR

This paper explores various categories of graphs with relaxed incidence and morphism conditions, analyzing their categorical properties and how they relate to Lawvere axioms and other categorical constructs.

Contribution

It introduces the Category of Conceptual Graphs with relaxed conditions and examines its categorical properties compared to standard graph categories.

Findings

The Category of Conceptual Graphs allows multiple edges between vertices.

Different graph categories are characterized by various restrictions on morphisms.

The applicability of Lawvere axioms varies across the different graph categories.

Abstract

In the standard Category of Graphs, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these graph morphisms as Strict Morphisms. We relax the condition on the graphs allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call this broader graph category The Category of Conceptual Graphs, and define four other graph categories created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs. We investigate which Lawvere axioms for the category of Sets and Functions apply to each of these Categories of Graphs, as well as the other categorial constructions of free objects, projective objects, generators, and their categorial duals.