Fernández Vilaboa, José ManuelGonzález Rodríguez, RamónRamos Pérez, Brais2025-06-232025-06-232025-05-14Fernández Vilaboa, J.M., González Rodríguez, R. & Ramos Pérez, B. Relative Rota-Baxter operators, modules and projections. Czech Math J (2025). https://doi.org/10.21136/CMJ.2025.0467-240011-4642https://hdl.handle.net/10347/42260The present article is devoted to introduce, in a braided monoidal setting, the notion of module over a relative Rota-Baxter operator. It is proved that there exists an adjunction between the category of modules associated to an invertible relative RotaBaxter operator and the category of modules associated to a Hopf brace, which induces an equivalence by assuming certain additional hypothesis. Moreover, the notion of projection between relative Rota-Baxter operators is defined, and it is proved that those which are called “strong” give rise to a module according to the previous definition in the cocommutative setting.engThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/Braided monoidal categoryHopf algebraHopf bracerelative Rota-Baxter operatorjournal article10.21136/CMJ.2025.0467-24open access