Algebraic models of cubical weak higher structures

Document Type : Research Paper

Author

Laboratoire de Mathématiques d’Orsay, UMR 8628 Université de Paris-Saclay and CNRS Bâtiment 307, Faculté des Sciences d’Orsay

Abstract

In this article we recast some of the results developped in articles [19, 22] but in the setup of cubical geometry. Thus we define a monad on ℂ𝕊ets whose algebras are models of cubical weak ∞-groupoids with connections. In addition, we define a monad on the category ℂ𝕊ets ×ℂ𝕊ets whose algebras are models of cubical weak ∞-functors, and a monad on the category ℂ𝕊ets ×ℂ𝕊ets ×ℂ𝕊ets ×ℂ𝕊ets whose algebras are models of cubical weak ∞-natural transformations.

Keywords

References

[1] Al-Agl, F.A., Brown, R., and Steiner, R., Multiple categories: The equivalence of a globular and a cubical approach, Adv. Math. 170 (2002), 71-118.
[2] Ara, D., “Sur les ∞-groupoïdes de Grothendieck et une variante ∞-catégorique”, Ph.D. Thesis, University Denis Diderot (Paris 7), 2010.
[3] Batanin, M., On the Penon method of weakening algebraic structures, J. Pure Appl. Algebra 172(1) (2002), 1-23.
[4] Batanin, M., Monoidal globular categories as a natural environment for the theory of weak-n-categories, Adv. Math. 136 (1998), 39-103.
[5] Bourke, J., Iterated algebraic injectivity and the faithfulness conjecture, https://arxiv.org/pdf/1811.09532.pdf.
[6] Brown, R., Higgins, Ph., and Sivera, R., “Nonabelian Algebraic Topology”, Europ. Math. Soc., Tracts in Mathematics 15, 2011.
[7] Brown, R., A new higher homotopy groupoid: the fundamental globular ω-groupoid of a filtered space, Homology Homotopy Appl. 10(1) (2008), 327-343.
[8] Brown, R., Double modules, double categories and groupoids, and a new homotopy double groupoid, arXiv Math (0903.2627) (2009), 7 pages.
[9] Brown, R. and Higgins, Ph., The equivalence of ∞-groupoids and crossed complexes, Cah. Topol. Géom. Différ. 22(4) (1981), 371-386.
[10] Brown, R. and Higgins, Ph., The equivalence of ω-groupoids and cubical T-complexes, Cah. Topol. Géom. Différ. 22(4) (1981), 349-370.
[11] Brown, R. and Higgins, Ph., Tensor products and homotopies for ω-groupoids and crossed complexes, J. Pure Appl. Algebra 47(1) (1987), 1-33.
[12] Brown, R. and Loday, J.L., Homotopical excision, and Hurewicz theorems for n-cubes of spaces, Proc. London Math. Soc. 54(1) (1987), 176-192.
[13] Brown, R. and Spencer, C.B.,Double groupoids and crossed modules, Cah. Topol. Géom. Différ. 17(4) (1976), 343-362.
[14] Coppey, L. and Lair, Ch. Leçons de théorie des esquisses, Université Paris VII, (1985).
[15] Ehresmann, Ch, Problémes universels relatifs aux catégories n-aires, C. R. Math. Acad. Sci. Soc. 264 (1967), 273-276.
[16] Feshbach, M. and Voronov, A., A higher category of cobordisms and topological quantum field theory, https://arxiv.org/abs/1108.3349.
[17] Foltz, F., Sur la catégorie des foncteurs dominés, Cah. Topol. Géom. Différ. Catég. 11(2), (1969), 101-130.
[18] Grandis, M. and Paré, R., An introduction to multiple categories (On weak and lax multiple categories, I), Cah. Topol. Géom. Différ. Catég. LVII(2) (2016), 103-159.
[19] Kachour, K., Définition algébrique des cellules non-strictes, Cah. Topol. Géom. Différ. Catég. 1 (2008), 1-68.
[20] Kachour, C., (∞, n)-ensembles cubiques, talk in CLE, Paris, November 2016, https: //sites.google.com/site/logiquecategorique/Contenus/201611-kachour.
[21] Kachour, C., Algebraic Models of Cubical Weak ∞-Categories with Connections, To appear in Categ. General Alg. Structures Appl.
[22] Kachour, C., Algebraic definition of weak (∞, n)-categories, Theory Appl. Categ. 30(22) (2015), 775-807.
[23] Kachour, C., An algebraic approach to weak ω-groupoids, Australian Category Seminar, 14 September 2011, http://web.science.mq.edu.au/groups/coact/seminar/ cgi-bin/speaker-info.cgi?name=Camell+Kachour.
[24] Kachour, C. and Weber, M., “Introduction to Higher Cubical Operads”, First Part: The Cubical Operad of Cubical Weak ∞-Categories, Prépublication de l’IHES, http://preprints.ihes.fr/2017/M/M-17-19.pdf.
[25] Kachour, C. and Weber, M., “Introduction to Higher Cubical Operads”, Second Part: The Functor of Fundamental Cubical Weak ∞-Groupoids for Spaces, Prépublication de l’IHES, http://preprints.ihes.fr/2018/M/M-18-03.pdf.
[26] Lair, Ch., “Condition syntaxique de triplabilité d’un foncteur algébrique esquissé”, Diagrammes 1, 1979.
[27] Lucas, M., Cubical (ω, p)-categories, High. Struct. 2 (2018), 191-233.
[28] Lurie, J., On the classification of topological field theories (draft), https://arxiv. org/pdf/0905.0465.pdf.
[29] Maltsiniotis, G., La catégorie cubique avec connections est une catégorie test stricte, (preprint) (2009), 1-16.
[30] Maltsiniotis, G., Grothendieck ∞-groupoids, and still another definition of ∞-catégories, Available online (2010): http://arxiv.org/pdf/1009.2331v1.pdf.
[31] Penon, J., Approche polygraphique des ∞-catLegories non-strictes, Cah. Topol. GLeom. DiffLer. CatLeg. (1999), 31-80.
[32] Penon, J., ∞-catégorification de structures équationnelles, Séminaires Itinérants de Catégories (S.I.C) á Amiens, Septembre 2005.
Categories and General Algebraic Structures with Applications
Volume 16, Issue 1
January 2022
Pages 189-220

Files

Share

How to cite

Statistics