Computational Paths -- A Weak Groupoid

Abstract

We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as our tool, to perform in algebraic topology an approach of computational paths. This makes it possible to build the fundamental groupoid of a type $X$ connected by paths. Then, we will establish the morphism between these groupoid structures, getting the concept of isomorphisms between types and to constitute the category of computational paths, which will be called $\mathcal{C}_{paths}$. Finally, we will conclude that the weak category $\mathcal{C}_{paths}$ determines a weak groupid.Comment: 37 pages. arXiv admin note: substantial text overlap with arXiv:1906.0910