On the Groupoid Model of Computational Paths

Abstract

The main objective of this work is to study mathematical properties of computational paths. Originally proposed by de Queiroz \& Gabbay (1994) as `sequences or rewrites', computational paths are taken to be terms of the identity type of Martin L\"of's Intensional Type Theory, since these paths can be seen as the grounds on which the propositional equality between two computational objects stand. From this perspective, this work aims to show that one of the properties of the identity type is present on computational paths. We are referring to the fact that that the identity type induces a groupoid structure, as proposed by Hofmann \& Streicher (1994). Using categorical semantics, we show that computational paths induce a groupoid structure. We also show that computational paths are capable of inducing higher categorical structures.