Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.1511335 Category Theory, K Theory, Homological Algebra Kategori Teorisi, K Teorisi, Homolojik Cebir Categorical isomorphisms for Hopf braces https://orcid.org/0000-0002-5995-7961 Fernadez Vılaboa J.m. University of Santiago de Compostela https://orcid.org/0000-0003-3061-6685 Gonzalez Rodriguez Ramon University of Vigo https://orcid.org/0009-0006-3912-4483 Ramos Pérez Brais University of Santiago de Compostela 10 29 2025 54 5 1872 1896 07 22 2024 02 19 2025 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

In this paper, we introduce the category of brace triples in a braided monoidal setting and prove that it is isomorphic to the category of s-Hopf braces, which are a generalization of cocommutative Hopf braces. After that, we obtain a categorical isomorphism between the category of finite cocommutative Hopf braces and a certain subcategory of the category of cocommutative post-Hopf algebras, which supposes an expansion to the braided monoidal setting of the equivalence obtained for the category of vector spaces over a field $\mathbb{K}$ by Y. Li, Y. Sheng and R. Tang.

 Braided monoidal category Hopf algebra Hopf brace brace triple post-Hopf algebra PID2020-115155GB-I00 [1] J.N. Alonso Álvarez, J.M. Fernández Vilaboa and R. González Rodríguez, On the (co)-commutativity class of a Hopf algebra and crossed products in a braided category, Comm. Algebra 29 (12), 5857-5878, 2001. [2] I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang-Baxter operators, Proc. Amer. Math. Soc. 145 (5), 1981-1995, 2017. [3] C. Bai, L. Guo, Y. Sheng and R. Tang, Post-groups, (Lie-)Butcher groups and the YangBaxter equation, Math. Ann. 388 (3), 1-41, 2024. [4] R.J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1), 193-228, 1972. [5] V.G. Drinfeld, On some unsolved problems in quantum group theory, in: Quantum groups, Leningrad, 1990, 1-8, Springer, Berlin, 1992. [6] J.M. Fernández Vilaboa, R. González Rodríguez, B. Ramos Pérez and A.B. Rodríguez Raposo, Modules for invertible 1-cocycles, Turkish J. Math. 48 (2), 248-266, 2024. [7] R. González Rodríguez, The fundamental theorem of Hopf modules for Hopf braces, Linear Multilinear Algebra 70 (20), 5146-5156, 2022. [8] R. González Rodríguez and A.B. Rodríguez Raposo, Categorical equivalences for Hopf trusses and their modules, arXiv: 2312.06520 [math.RA]. [9] L. Guarnieri and L. Vendramin, Skew braces and the YangBaxter equation, Math. Comp. 86 (307), 2519-2534, 2017. [10] J.A. Guccione, J.J. Guccione and L. Vendramin, YangBaxter operators in symmetric categories, Comm. Algebra 46 (7), 2811-2845, 2018. [11] A. Joyal and R. Street, Braided monoidal categories, Macquarie Univ. Reports 860081, 1986. [12] A. Joyal and R. Street, Braided tensor categories, Adv. Math. 102 (1), 20-78, 1993. [13] C. Kassel, Quantum Groups, Springer-Verlag, 1995. [14] Y. Li, Y. Sheng and R. Tang, Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation, J. Noncommut. Geom. 18 (2), 605-630, 2024. [15] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, 1998. [16] S. Majid, Transmutation Theory and Rank for Quantum Braided Groups, Math. Proc. Cambridge Philos. Soc. 113 (1), 45-70, 1993. [17] W. Rump, Braces, radical rings, and the quantum YangBaxter equation, J. Algebra 307 (1), 153-170, 2007. [18] P. Schauenburg, On the braiding on a Hopf algebra in a braided category, New York J. Math. 4, 259-263, 1998. [19] C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (23), 1312-1315, 1967.