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arXiv · 2205.08952

Zigzag normalisation for associative $n$-categories

Lukas Heidemann·David Reutter·Jamie Vicary
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Abstract

The theory of associative $n$-categories has recently been proposed as a strictly associative and unital approach to higher category theory. As a foundation for a proof assistant, this is potentially attractive, since it has the potential to allow simple formal proofs of complex high-dimensional algebraic phenomena. However, the theory relies on an implicit term normalisation procedure to recognize correct composites, with no recursive method available for computing it. Here we describe a new approach to term normalisation in associative $n$-categories, based on the categorical zigzag construction. This radically simplifies the theory, and yields a recursive algorithm for normalisation, which we prove is correct. Our use of categorical lifting properties allows us to give efficient proofs of our results. This normalisation algorithm forms a core component of the proof assistant homotopy.io, and we illustrate our scheme with worked examples.

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Lukas Heidemann, David Reutter, Jamie Vicary. 2022-05-18. Zigzag normalisation for associative $n$-categories. https://doi.org/10.1145/3531130.3533352

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