Path and Boundary-Path Groupoids

• Path and boundary-path groupoids are mathematical structures that encode both finite combinatorial paths and their asymptotic dynamics across directed graphs and higher categories.
• They enable the construction of C*-algebras, symplectic groupoids, and invariants in noncommutative geometry, directly impacting operator algebra and quantum field theory.
• Multiple frameworks, such as filter models, graph morphisms, and tight groupoid constructions, yield isomorphic groupoids that unify dynamical, topological, and algebraic analyses.

A path groupoid encodes the combinatorial and topological data of paths—typically finite or infinite—in a directed combinatorial structure such as a graph, higher-rank graph ($k$-graph), or, more generally, a category of paths. The boundary-path groupoid is a refinement that encodes the limiting, or "boundary," dynamics: in graph-theoretic settings, this is often the dynamics at infinity of paths, and in higher categorical, homotopical, or geometric contexts, it formalizes the asymptotic structure, completions, or moduli of paths up to various equivalence relations. These groupoids support the construction of important operator algebras, model dynamics in Poisson geometry, give rise to invariants in noncommutative geometry, and unify perspectives across topology, category theory, and mathematical physics.

1. Definitions and Core Frameworks

The path groupoid for a directed graph $E$ or a higher-rank graph $\Lambda$ is typically constructed as follows:

• Objects (Units): The space of infinite paths, or, more generally, a space of directed hereditary subsets of the generating category, often called the path space.
• Morphisms: Elements correspond to triples $(x, k, y)$, where $x$, $y$ are infinite (or boundary) paths and $k$ is a "lag" parameter, subject to the existence of a finite initial segment or shift equivalence (e.g., there exist finite paths $\alpha$, $\beta$ and tail $z$ such that $x = \alpha z$, $y = \beta z$, and $|\alpha| - |\beta| = k$) (Rigby, 2018, Carlsen et al., 2016).
• Boundary-path groupoid: Restricts attention to certain "boundary" (often, maximal or aperiodic) paths: these play the role of "aperiodic" or "maximal" points in the underlying path space, and yield the boundary or tight groupoid in semigroup language (Webster, 2011, Castro et al., 2021).

In higher-rank and non-finitely aligned settings, the path space is constructed using hereditary, directed filters whose basic, or "cylinder," sets are defined in terms of initial segments or directed hereditary categories (Jones, 26 Mar 2025, Clark et al., 17 Sep 2025). In the general categorical case, the groupoid elements reflect zigzag relations or equivalence classes of concatenations and cancellations (Spielberg, 2011, Spielberg, 2017).

Fundamental Topological Structure

• Locally compact Hausdorff path spaces are constructed using cylinder sets, often of the form $Z(p) = \{x : x(0,d(p)) = p\}$ and their set-theoretic differences.
• Boundary-path space $\partial \Lambda$ is typically a closed subset, defined by regularity or aperiodicity constraints, or as the closure of maximal filters (Webster, 2011, Jones, 26 Mar 2025).
• Unit space: For ample groupoids, this is totally disconnected (zero-dimensional), and the groupoid is étale.

2. Generalization and Multi-Perspective Models

The construction of path and boundary-path groupoids has been extensively generalized:

Approach	Applicable Setting	Key Features
Filter Models	Finitely aligned P-graphs/Cats	Directed hereditary filters
Graph Morphisms	k-graphs, P-graphs	Functors from path prototypes
Inverse Semigroup/Tight	(Generalized) Boolean systems, LCSC	Tight spectrum, Exel's groupoids

• The filter approach models path and boundary-path spaces as sets of directed hereditary subsets, with a topology determined by pointwise convergence (Clark et al., 17 Sep 2025).
• Graph morphism models employ functors from finite prototypes (e.g., $\Omega_{P,(m_n)}$) into the combinatorial category, generalizing the standard $\Omega_{k,m}$ model for $k$-graphs (Clark et al., 17 Sep 2025).
• The tight groupoid construction, critical in inverse semigroup $C^*$-algebra theory, uses the tight spectrum of idempotents and connects with boundary-path models in Boolean dynamical systems (Castro et al., 2021).

The central result is that, for finitely aligned higher-rank graphs (extending to finitely aligned P-graphs), all these approaches yield isomorphic groupoids—established via explicit conjugacies of semigroup actions and groupoid isomorphisms (Clark et al., 17 Sep 2025).

3. Dynamical and Operator Algebraic Implications

Path and boundary-path groupoids are crucial in constructing operator algebras and analyzing their properties:

• Groupoid $C^*$-algebras: Every path groupoid yields a locally compact, ample (often Hausdorff) étale groupoid whose groupoid $C^*$-algebra encodes the dynamics and combinatorics of the original system (Webster, 2011, Rigby, 2018, Spielberg, 2011).
• Diagonal subalgebras and spectra: The boundary-path space $\partial\Lambda$ is homeomorphic to the spectrum of the commutative subalgebra $D_{\Lambda}$ generated by range projections. This aligns the topological structure of the boundary-path groupoid with the representation theory of $C^*(\Lambda)$ (Webster, 2011).
• Uniqueness theorems: The Cuntz-Krieger and gauge-invariant uniqueness theorems are proved by analyzing topological freeness (aperiodicity) and the ideal intersection properties afforded by the groupoid model (Spielberg, 2011, Hazlewood, 2013).

When a higher-rank graph $\Lambda$ is row-finite, $C^*(\Lambda)$ is often realized as a full corner in the $C^*$-algebra of a desourcified graph $\tilde{\Lambda}$, transferring topological and algebraic structure (e.g., the spectral theory of the diagonal) (Webster, 2011). For amenable groupoids, nuclearity and AF core results are deduced (Spielberg, 2011, Jones, 26 Mar 2025).

For Leavitt path algebras, the Steinberg algebra of the boundary-path groupoid is (graded) isomorphic to $L_R(E)$, with correspondence of grading and explicit isomorphisms on generators and relations (Rigby, 2018).

4. Advanced Generalizations and Connections

Non-Finitely Aligned and Local Finite Alignment

• The path groupoid construction is extended to non-finitely aligned $k$-graphs by identifying the finitely aligned part $\mathrm{FA}_\Lambda$ and restricting filters to those intersecting $\mathrm{FA}_\Lambda$, ensuring local compactness; groupoids constructed from such path spaces are ample and Hausdorff (Jones, 26 Mar 2025).
• Inverse semigroup models (or tight groupoids) for boundary-path spaces generalize to generalized Boolean dynamical systems (Castro et al., 2021), and similar extensions apply for left-cancellative small categories (Spielberg, 2017).

Homotopical and Higher Categorical Models

• In the homotopy theory of quasi-categories, the path groupoid and the system of "boundary-path groupoids" (cores of mapping complexes) replace classical higher homotopy groups. Weak equivalences induce equivalences on these groupoids, providing a robust invariant for higher category theory (Jardine, 2019).
• In path categories enriched over groupoids, path and boundary-path groupoids encode morphisms and homotopies (as 1- and 2-morphisms), realizing internal homotopy-theoretic types (homotopy exponentials and $\Pi$-types) (Besten, 2020).

Geometric, Quantum, and Field-Theoretic Contexts

• In Poisson geometry and the theory of Poisson sigma models, path and boundary-path groupoids (and their infinite-dimensional or relational symplectic avatars) model the phase space and reduction by gauge symmetries, leading to symplectic groupoids. These constructions are central to deformation quantization and higher gauge theory (Contreras et al., 2012, Contreras, 2013, Lackman, 2023).
• Higher path groupoids (e.g., 2- and 3-groupoids of manifolds) are the natural domains for higher holonomy functors associated to formal power series connections, with applications to configurations spaces, knot invariants, and topological quantum field theory (Cellot, 30 Jun 2025).

5. Simplicity, Ideals, and Dynamical Invariants

Path and boundary-path groupoids play a fundamental role in the simplicity and ideal structure of associated algebras:

• C*-simplicity criterion: For minimal groupoids, the absence of recurrent amenable isotropy subgroups ensures the intersection property and C*-simplicity, with the Furstenberg boundary (spectrum of the groupoid-equivariant injective envelope) playing the role of detecting trivial isotropy (Borys, 2019).
• Orbit equivalence and groupoid isomorphisms: In graph theory, a homeomorphism of boundary path spaces that respects isolated eventually periodic points leads to isomorphic graph groupoids, controlling the invariance of operator algebraic structure under orbit dynamics (Carlsen et al., 2016).
• Structural consequences: For classifying liminal, postliminal, Cartan, and Fell C*-algebras, the principal properties of the path groupoid and the nature of boundary-path spaces (closedness of orbits, uniqueness up to shift, etc.) are decisive (Hazlewood, 2013).

6. Comparison and Unification of Different Models

For finitely aligned higher-rank graphs (and generalizations), multiple constructions of the path and boundary-path groupoids are now known to yield topologically isomorphic groupoids, including:

• The filter and graph morphism models (Clark et al., 17 Sep 2025)
• Spielberg's groupoid constructions (Jones, 26 Mar 2025, Clark et al., 17 Sep 2025, Spielberg, 2011)
• Tight groupoid of an associated inverse semigroup (Castro et al., 2021, Clark et al., 17 Sep 2025)
• Renault–Williams semidirect product groupoids in the presence of suitable semigroup actions (Jones, 26 Mar 2025)

These equivalences hold at the level of both the path groupoid and its boundary restriction, facilitating a unified approach to operator algebraic, combinatorial, and dynamical analysis.

7. Applications and Further Directions

• Quantum field theory and symplectic geometry: Path and boundary-path groupoids underpin the construction of phase spaces in Poisson sigma models, the categorical structure of quantization, and higher holonomy invariants (Contreras et al., 2012, Contreras, 2013, Lackman, 2023, Cellot, 30 Jun 2025).
• Computational and type-theoretic frameworks: In intensional type theory, the identity type gives rise to a weak groupoid structure on computational paths, with further generalizations to weak $\omega$-groupoids and higher categorical structure (Ramos et al., 2015, Besten, 2020).
• Geometric group theory: Visual boundaries of groups with isolated flats are path-connected, providing a topological setting for defining path groupoids at infinity, relevant for semistability and rigidity at infinity (Ben-Zvi, 2019).

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In summary, the theory of path and boundary-path groupoids provides a versatile and unifying language for discrete dynamics, operator algebra, noncommutative topology, higher category theory, and mathematical physics. Across combinatorial, homotopical, and analytical domains, the groupoid approach both organizes computations and reveals deep structural invariants.