A rewriting coherence theorem with applications in homotopy type theory

Kraus, Nicolai; von Raumer, Jakob

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 by laying out the construction in the language of higher-dimensional rewriting.

Citation

Kraus, N., & von Raumer, J. (2022). A rewriting coherence theorem with applications in homotopy type theory. Mathematical Structures in Computer Science, 32(7), 982-1014. https://doi.org/10.1017/s0960129523000026

• Journal Article Type: Article
• Acceptance Date: Jan 6, 2023
• Online Publication Date: Feb 6, 2023
• Publication Date: Aug 6, 2022
• Deposit Date: Mar 23, 2023
• Publicly Available Date: Mar 24, 2023
• Journal: Mathematical Structures in Computer Science
• Print ISSN: 0960-1295
• Electronic ISSN: 1469-8072
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 32
• Issue: 7
• Pages: 982-1014
• DOI: https://doi.org/10.1017/s0960129523000026
• Keywords: Homotopy type theory; higher-dimensional rewriting; constructive mathematics; coherence of structure; well- founded relation; confluence; Squier theory; abstract rewriting system; polygraph
• Public URL: https://nottingham-repository.worktribe.com/output/18814063
• Publisher URL: https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/on-reduction-and-normalization-in-the-computational-core/14DE21ED5F5B6A62EDD17E210D3A41C5