🤖 AI Summary
This work addresses the challenge in semantic modeling of scoped computational paths, where algebraic structure and topological coherence are difficult to reconcile due to incompatibility between product and quotient topologies. By constructing a topological semantics for scoped rewriting systems, it endows rewrite steps with continuous geometric realizations and characterizes name-carrying rewrites via endpoint-fixing homotopies. The paper introduces an unconditional construction of a final composable topology that explicitly resolves—rather than obscures—the product-quotient inconsistency. It establishes quadruple equivalence criteria and sufficient conditions for compact Hausdorffness, and proves, via continuous sections in a universal representation, that the coherent path quotient is homeomorphic to the standard fundamental groupoid. Formal verification in Lean 4.24.0 covers finitely generated cases such as the circle and torus, yielding normal forms and ℤ, ℤ² classifications via winding numbers, and confirming that the realization map is a continuous groupoid homomorphism and faithful under geometric completeness.
📝 Abstract
Computational paths record equality as explicit finite traces of primitive steps. We give a topological semantics for a scoped rewrite presentation whose steps have continuous geometric realizations and whose named rewrites carry endpoint-fixed homotopies.
For every presentation we construct a quotient arrow space with a canonical final-domain groupoid structure: multiplication is continuous on the quotient of explicitly composable representatives. We prove an exact four-way criterion for this final composable topology to agree with the ordinary pullback topology, together with a compact-Hausdorff sufficient condition. Thus the unconditional construction exposes, rather than hides, the product-quotient issue in ordinary topological groupoids.
The realization map to geometric homotopy classes is a continuous groupoid morphism and is faithful exactly under a separate geometric-completeness condition. In the universal presentation, a continuous section identifies the coherent-path quotient homeomorphically with the usual quotient-topologized fundamental groupoid. We then give finite-generator circle and genuine torus examples, with winding-based normal forms and classifications by Z and Z^2. A Lean 4.24.0 development checks the theorem package; the mathematical presentation is independent of the implementation.