Involutive Weak Cubical ω-categories
Main Article Content
Abstract
We investigate the notion of involutive weak cubical ω-categories via Penon’s approach: as algebras for the monad induced by the free involutive strict ω-category functor on cubical ω-sets. A few examples of involutive weak cubical ω-categories are provided.
Article Details

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
References
Eilenberg S, Mac Lane S. General Theory of Natural Equivalences. Trans Am Math Soc. 1954;58:231-94.
Ehresmann C. Catégories et Structures Dunod (1965), Paris.
Roberts J.E. Mathematical Aspects of Local Cohomology. Algébres d’Opérateurs et Leurs Applications en Physique Mathématique.
Brown R, Higgins P. Sur les Complexes Croisés ω-grupoides et T-complexes. C R Acad Sci Paris A. 1977;285:997-9.
Bénabou J Introduction to Bicategories. In: Reports of the Midwest Category Seminar, Lecture Notes in Math 47, Springer-Verlag, Berlin, 1967;1-77.
Street R. The Formal Theory of Monads. Journal of Pure and Applied Algebra. 1972;2(2):149-68.
Cheng E, Lauda A. Higher-Dimensional Categories: an Illustrated Guide Book. IMA Workshop. 2004.
Leinster T. Structures in Higherdimensional Category Theory. 2001.
Leinster T. Higher Operads, Higher Categories. Cambridge University Press. 2004.
Batanin M. Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-categories. Advances in Mathematics. 1998;136(1):39-103.
Penon J. Approche Polygraphique des ω-Categories Non Strictes. Cahiers de Topologie et Géometrie Différentielle. 1999;40(1):31-80.
Kachour C. Algebraic Models of Cubical Weak 1-categories with Connections. Categ Gen Algebr Struct Appl. 2022;16(1):143-87.
Lect.Notes Phys. 813 (2010), 95-172, Physics, topology, logic and compIn New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95-174.
Bertozzini P, Conti R, Lewkeeratiyutkul W, Suthichitranont N. On Strict Higher C*-categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques. 2020;LXI(3):239-348.
Yau D. Involutive category theory. (English) Zbl 07283178 Lecture Notes in Mathematics 2279. Cham: Springer. xii, 243 p. (2020).
Ghez P, Lima R, Roberts J.E. W*-categories. Pacific J Math. 1985;120(1):79-109.
Mitchener P. C*-categories. Proc London Math Soc. 2002;84:375-404.
Selinger P. (2005) Dagger Compact Closed Categories and Completely Positive Maps. Proceedings of the 3rd International Workshop on Quantum Programming Languages (Chicago June 30-July 1).
Abramsky S, Coecke B. (2004) A Categorical Semantics of Quantum Protocols. Proceedings of the 19th IEEE Conference on Logic in Computer Science (LiCS04).
Bejrakarbum P. (2016) Involutive Weak Globular Higher Categories. MSc Thesis Thammasat University.
Bejrakarbum P, Bertozzini P. Involutive weak globular higher categories. Proceedings of the 22nd Annual Meeting in Mathematics (AMM 2017), Chiang Mai University. 2017.
Bejrakarbum P. (2023) Involutive Weak Globular Higher Categories: Leinster’s Approach. PhD Thesis Thammasat University.
Bejrakarbum, P, ertozzini, P. (2023). Involutive Weak Globular ω-categories.
Bertozzini P, Conti R, Dawe Martins R (2014) Involutive Double Categories manuscript [BJ]. Jan 2013; 15-24.
Leinster T. (2014) Basic Category Theory. Cambridge University. Press.
Riehl E. (2016) Category Theory in Context. Dover.
Bertozzini P, Conti R, Puttirungroj C. Dualities for Multimodules. Indagationes Mathematicae. 2022;33:768-800.
Al-Agl F.A, Brown R, Steiner R. Multiple Categories: the Equivalence of a Globular and a Cubical Approach. 2002;170(1):71-118.