In this paper we introduce the notion of generalized invertible 1-cocycle in a strict braided monoidal category \(\textsf{C}\) , 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 \(\textsf{C}\) 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.
