DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Thin path groupoid

Definition

A groupoid is a category in which every morphism has a two-sided inverse. For an open subset $U$ of a smooth manifold, the thin path groupoid $Π_{1}^{t h i n} (U)$ is the category whose objects are the points of $U$ and whose morphisms from $x$ to $y$ are tagged triples $(x, y, [γ])$ , where $[γ]$ is a thin-homotopy class of piecewise-smooth paths from $x$ to $y$ ; composition, identities, and inverses are those of Proposition Thin path classes form a groupoid.

Definition references

Piecewise-smooth paths and thin homotopies

Thin path classes form a groupoid

Equivalence relation, equivalence class, and the quotient set $A / \sim$

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Depends on

Dependency tree · 2 levels

Sources

Urs Schreiber and Konrad Waldorf, Parallel Transport and Functors, §2.1, Definition 2.4