A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

TL;DR

This paper develops a method to construct homotopy bases for higher-dimensional rewriting systems using confluence and wellfoundedness, and applies it to problems in homotopy type theory, advancing the understanding of higher coherence structures.

Contribution

It introduces a recursive construction of homotopy bases from confluence and wellfoundedness, translating it into homotopy type theory and addressing open questions in the field.

Findings

Constructs homotopy bases using confluence and wellfoundedness.

Applies the construction to homotopy type theory to manage higher equalities.

Provides insights into homotopy groups of free groups and pushouts of 1-types.

Abstract

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.