Arthur Freitas Ramos | Searcharxiv

arXiv2026

Topological Semantics for Scoped Computational Paths

Computational paths record the steps of an equality derivation. We give them a topological semantics that distinguishes derivable rewrites from arbitrary homotopies. Coherent representatives pair traces with paths homotopic to their realizations. We compare a topology retaining the entire trace with one observing only endpoints, length, and paths. Quotienting by the declared rewrites gives a groupoid. Multiplication is continuous when composable pairs carry the quotient topology inherited from composable representatives. This topology can differ from the usual subspace topology on pairs of quotient arrows. We characterize when they agree, give compact-Hausdorff and discrete sufficient conditions, and use the Hawaiian earring to exhibit a failure of agreement. The comparison with geometric homotopy classes is injective exactly when the presentation is geometrically complete. Normal-form certificates give a criterion for completeness. In the universal presentation, all paths are primitive steps and all endpoint-fixed homotopies are allowed rewrites; its quotient recovers the quotient-topologized fundamental groupoid. Circle and torus examples recover the classical based-loop classifications by $\mathbb Z$ and $\mathbb Z^2$. A Lean development supports the construction. A focused Lean 4.32.0 result registered in Palomar covers the topology comparison, additive circle and torus classifications, and a conditional Hawaiian-earring obstruction transfer. We distinguish that result from the earlier Lean 4.24.0 development and from the mathematical exposition.

cs.LO↗

arXiv2018

Explicit Computational Paths

The treatment of equality as a type in type theory gives rise to an interesting type-theoretic structure known as `identity type'. The idea is that, given terms $a,b$ of a type $A$, one may form the type $Id_{A}(a,b)$, whose elements are proofs that $a$ and $b$ are equal elements of type $A$. A term of this type, $p : Id_{A}(a,b)$, makes up for the grounds (or proof) that establishes that $a$ is indeed equal to $b$. Based on that, a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a `computational path'. One interesting fact is that it is possible to rewrite computational paths using a set of reduction rules arising from an analysis of redundancies in paths. These rules were mapped by De Oliveira in 1994 in a term rewrite system known as $LND_{EQ}-TRS$. Here we use computational paths and this term rewrite system to develop the main foundations of homotopy type theory, i.e., we develop the lemmas and theorems connected to the main types of this theory, types such as products, coproducts, identity type, transport and many others. We also show that it is possible to directly construct path spaces through computational paths. To show this, we construct the natural numbers and the fundamental group of the circle, showing results connected to these structures.

cs.LO↗