Sequences of Rewrites: A Categorical Interpretation

TL;DR

This paper provides a categorical interpretation of sequences of rewrites in type theory, simplifying the understanding of identity types and connecting them to known mathematical structures like groupoids.

Contribution

It introduces a categorical framework for sequences of rewrites in type theory, aligning with existing structures and extending the interpretation to complex rewrite sequences.

Findings

Types form a groupoidal structure under this interpretation

Sequences of rewrites can be modeled as morphisms in a category

The interpretation aligns with known results in type and homotopy theory

Abstract

In Martin-L\"of's Intensional Type Theory, identity type is a heavily used and studied concept. The reason for that is the fact that it's responsible for the recently discovered connection between Type Theory and Homotopy Theory. The main problem with identity types, as originally formulated, is that they are complex to understand and use. Using that fact as motivation, a much simpler formulation for the identity type was proposed by Queiroz & Gabbay (1994) and further developed by de Queiroz & de Oliveira (2013). In this formulation, an element of an identity type is seen as a sequence of rewrites (or computational paths). Together with the logical rules of this new entity, there exists a system of reduction rules between sequence of rewrites called LND_{EQS}-RWS. This system is constructed using the labelled natural deduction (i.e. Prawitz' Natural Deduction plus derivations-as-terms) and is responsible for establishing how a sequence of rewrites can be rewritten, resulting in a new sequence of rewrites. In this context, we propose a categorical interpretation for this new entity, using the types as objects and the rules of rewrites as morphisms. Moreover, we show that our interpretation is in accordance with some known results, like that types have a groupoidal structure. We also interpret more complicated structures, like the one formed by a rewrite of a sequence of rewrites.