Groupoids in combinatorics -- applications of a theory of local symmetries

TL;DR

This paper explores the application of groupoid theory to combinatorics, introducing concepts like holonomy and curvature to analyze structures such as graphs and polytopes, and discusses potential applications and open problems.

Contribution

It provides a unified exposition of combinatorial groupoid concepts and highlights their potential applications across various combinatorial and geometric structures.

Findings

Introduction of holonomy and curvature in combinatorial contexts

Unified framework for combinatorial groupoids

Identification of open problems and future research directions

Abstract

An objective of the theory of combinatorial groupoids is to introduce concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature" etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes, arrangements and other combinatorial objects. In this paper we give an exposition of some of the currently most active research themes in this area, offer a unified point of view, and provide a list of prospective applications in other fields together with a collection of related open problems.