Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T10:57:09.900Z Has data issue: false hasContentIssue false
  • English
  • Français

A cubical Squier’s theorem

Published online by Cambridge University Press:  06 February 2020

Maxime Lucas Open the ORCID record for Maxime Lucas [Opens in a new window]
Maxime Lucas*
Affiliation:
Inria, LS2N, Université de Nantes, France
*
Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 Guiraud and Malbos, 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.

Keywords

Cubical categoriescoherencerewriting
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 2 , February 2020 , pp. 159 - 172
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al-Agl, F. A., Brown, R. and Steiner, R. (2002). Multiple categories: The equivalence of a globular and a cubical approach. Advances in Mathematics 170 (1), 71–118. doi:10.1006/aima.2001.2069.CrossRefGoogle Scholar
Batanin, M. A. (1998). Computads for finitary monads on globular sets. In: Higher Category Theory (Evanston, IL, 1997), Vol. 230, 37–57. Contemp. Math. Amer. Math. Soc. doi:10.1090/conm/230/03337.CrossRefGoogle Scholar
Brown, R. and Higgins, P. J. (1977). Sur les complexes croisés, ω-groupoïdes, et T-complexes. C. R. Acad. Sci. Paris Sér. A-B 285 (16), A997–A999.Google Scholar
Brown, R. and Higgins, P. J. (1981). On the algebra of cubes. Journal of Pure and Applied Algebra 21 (3), 233–260. doi:10.1016/0022-4049(81)90018–9.CrossRefGoogle Scholar
Brown, R. and Mosa, G. H. (1999). Double categories, 2-categories, thin structures and connections. Theory and Applications of Categories, 5 (7), 163–175.Google Scholar
Burroni, A. (1993). Higher-dimensional word problems with applications to equational logic. In: 4th Summer Conference on Category Theory and Computer Science (Paris, 1991), Vol. 115, 1, pp. 43–62. doi:10.1016/0304-3975(93)90054-W.CrossRefGoogle Scholar
Garner, R. (2010). Homomorphisms of higher categories. Advances in Mathematics 224 (6), 2269–2311. doi:10.1016/jaim.2010.01.022.CrossRefGoogle Scholar
Guiraud, Y. and Malbos, P. (2012a). Coherence in monoidal track categories. Mathematical Structures in Computer Science 22 (6), 931–969. doi:10.1017/S096012951100065X.CrossRefGoogle Scholar
Guiraud, Y. and Malbos, P. (2012b). Higher-dimensional normalisation strategies for acyclicity. Advances in Mathematics 231 (3–4), 2294–2351.CrossRefGoogle Scholar
Guiraud, Y. and Malbos, P. (2018). Polygraphs of finite derivation type. Mathematical Structures in Computer Science 28 (2), 155–201. doi:10.1017/S0960129516000220.CrossRefGoogle Scholar
Higgins, P. J. (2005). Thin elements and commutative shells in cubical ω-categories. Theory and Applications of Categories, 14 (4), 60–74.Google Scholar
Lucas, M. (2017). A coherence theorem for pseudonatural transformations. Journal of Pure and Applied Algebra 221 (5), 1146–1217. doi:10.1016/j.jpaa.2016.09.005.CrossRefGoogle Scholar
Lucas, M. (2018). Cubical (ω,p)-categories. Higher Structures 2 (1), 191–233.Google Scholar
Squier, C. C. (1987). Word problems and a homological finiteness condition for monoids. Journal of Pure and Applied Algebra 49 (1–2), 201–217.CrossRefGoogle Scholar
Squier, C. C., Otto, F. and Kobayashi, Y. (1994). A finiteness condition for rewriting systems. Theoretical Computer Science 131 (2), 271–294.CrossRefGoogle Scholar
Street, R. (1976). Limits indexed by category-valued 2-functors. Journal of Pure and Applied Algebra 8(2), 149–181. doi:10.1016/0022-4049(76)90013-X.CrossRefGoogle Scholar
Street, R. (1987). The algebra of oriented simplexes. Journal of Pure and Applied Algebra 49 (3), 283–335. doi:10.1016/0022-4049(87)90137-X.CrossRefGoogle Scholar