Categorical equivalences for Hopf trusses and their modules

TL;DR

The paper establishes a braided-categorical bridge between Hopf trusses and generalized invertible 1-cocycles by proving an equivalence between the categories $\mathsf{HTr}$ and $\mathsf{GIC}$. It then develops module theories for both structures, showing that module categories correspond under this equivalence and extending known Hopf-brace results to the broader Hopf-truss framework. Under the assumption that $\mathsf{C}$ has equalizers, it defines left Hopf modules over a Hopf truss and proves a Fundamental Theorem of Hopf modules in this setting, yielding an equivalence between $H_1$-Hopf-Mod and $\mathbb{H}$-Hopf-Mod. Collectively, these results generalise invertible cocycle correspondences, module theory, and the Fundamental Theorem from Hopf braces to Hopf trusses, providing a cohesive, functorial framework in a braided monoidal context.

Abstract

In this paper we introduce the notion of generalized invertible 1-cocycle in a strict braided monoidal category C, and we prove that the category of Hopf trusses is equivalent to the category of generalized invertible 1-cocycles. On the other hand, we also introduce the notions of module for a Hopf truss and for a generalized invertible 1-cocycle. We prove some functorial results involving these categories of modules and we show that the category of modules associated to a generalized invertible 1-cocycle is equivalent to a category of modules associated to a suitable Hopf truss. Finally, assuming that in C we have equalizers, we introduce the notion of Hopf-module in the Hopf truss setting and we obtain the Fundamental Theorem of Hopf modules associated to a Hopf truss.

Theorems & Definitions (58)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Remark 2.9
• Remark 2.10