A unified categorical approach to graphs

TL;DR

This paper develops a unified coalgebraic framework for various graph types using endofunctors, extending universal coalgebra concepts to model graphs and their morphisms.

Contribution

It introduces a generalized coalgebraic approach to graphs via endofunctors, unifying different graph types under a common theoretical framework.

Findings

Graphs are modeled as 'co-like' structures sharing features of universal coalgebras.

Coalgebraic concepts like cofreeness and simulations are transferred to graph models.

Cofree constructions for graphs are less restrictive than traditional coalgebraic ones.

Abstract

For a set-endofunctor $F$, we extend the notion of universal $F$-coalgebras to $F$-graphs. These generalized coalgebras are models for various types of graphs, such as (un)directed (hyper)graphs, relational structures or fuzzy graphs. The induced morphisms coincide with graph homomorphisms. From this point of view, graphs are "co-like" structures and share features of universal coalgebras. In this article, we explore the coalgebraic character of graphs and transfer coalgebraic concepts like cofreeness, simulations or Co-Birkhoff theorems to $F$-graphs. Products and cofree constructions for $F$-graphs turn out to be less restrictive than their coalgebraic counterparts.