[Paper Review] Fundamental groupoids of k-graphs

David Pask, John Quigg+1 more|ArXiv.org|Jan 3, 2004

Advanced Operator Algebra ResearchMathematics12 references17 citations

TL;DR

This paper develops the fundamental groupoid of a k-graph using a universal construction from category theory, showing it as a quotient of the fundamental groupoid of the k-graph's 1-skeleton. The key contribution is a calculable framework for the fundamental groupoid despite the failure of k-graphs to embed faithfully into their groupoids, enabling combinatorial analysis via reduced-word techniques on the 1-skeleton.

ABSTRACT

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop a theory of the fundamental groupoid of a k-graph, and relate it to the fundamental groupoid of an associated graph called the 1-skeleton. We also explore the failure, in general, of k-graphs to faithfully embed into their fundamental groupoids.

Motivation & Objective

• To develop a theory of the fundamental groupoid for k-graphs, extending classical graph covering theory to higher-rank analogues.
• To address the failure of k-graphs to embed faithfully into their fundamental groupoids, which obstructs direct use of reduced-word arguments.
• To characterize the fundamental groupoid of a k-graph as a quotient of the fundamental groupoid of its 1-skeleton, enabling calculable and geometrically meaningful analysis.
• To lay foundational tools for a subsequent theory of coverings of k-graphs, as part of a broader program in higher-rank graph C*-algebras.

Proposed method

• Uses categories of fractions from category theory to construct the fundamental groupoid of a k-graph via a universal property.
• Represents the k-graph itself as a quotient of the path category of its 1-skeleton, using factorization relations among edges.
• Defines the fundamental groupoid of the k-graph as a quotient of the fundamental groupoid of the 1-skeleton, preserving structure under equivalence relations.
• Applies results from Schubert’s category theory and Bridson–Haefliger on fundamental groups of small categories to k-graphs.
• Establishes a geometric realization of the fundamental groupoid, showing agreement with the topological fundamental group of the associated space.
• Analyzes injectivity of the canonical functor from the k-graph to its fundamental groupoid, identifying obstructions and proposing alternatives.

Experimental results

Research questions

• RQ1 How can the fundamental groupoid of a k-graph be defined in a way that generalizes the classical graph case?
• RQ2 Why does the canonical functor from a k-graph to its fundamental groupoid fail to be injective, and what are the implications?
• RQ3 Can the fundamental groupoid of a k-graph be realized as a quotient of the fundamental groupoid of its 1-skeleton, and if so, how?
• RQ4 What is the relationship between the fundamental group of the k-graph and the fundamental group of its geometric realization?
• RQ5 Under what conditions can the image of the k-graph in its fundamental groupoid be used as a faithful replacement?

Key findings

• The fundamental groupoid of a k-graph is isomorphic to a quotient of the fundamental groupoid of its 1-skeleton, enabling calculations via reduced-word techniques.
• The canonical functor from the k-graph to its fundamental groupoid is not injective in general, as demonstrated by explicit counterexamples.
• The fundamental group of the k-graph agrees with the fundamental group of its geometric realization, validating the topological consistency of the construction.
• The image of the k-graph in its fundamental groupoid may serve as a replacement object, though it does not necessarily inherit the unique factorization property.
• The theory extends to arbitrary small categories, though the authors restrict to k-graphs due to lack of immediate applications.
• The construction provides a foundation for a theory of coverings of k-graphs, as the fundamental groupoid classifies such coverings.