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A cubical Squier’s theorem. (English) Zbl 1436.18022

Summary: The homotopical Squier’s theorem relates rewriting properties of a presentation of a monoid with homotopical invariants of this monoid. This theorem has since been extended by Y. Guiraud and P. Malbos [Math. Struct. Comput. Sci. 28, No. 2, 155–201 (2018; Zbl 1396.18004)], yielding a so-called polygraphic resolution of a monoid starting from a presentation with suitable rewriting properties. In this article, we argue that cubical categories are a more natural setting in which to express and possibly extend Guiraud and Malbos construction. As a proof-of-concept, we give a new proof of Squier’s homotopical theorem using cubical categories.

MSC:

18N30 Strict omega-categories, computads, polygraphs
55U35 Abstract and axiomatic homotopy theory in algebraic topology
20N02 Sets with a single binary operation (groupoids)
68Q42 Grammars and rewriting systems

Citations:

Zbl 1396.18004
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References:

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