Topological Semantics for Scoped Computational Paths

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.