Lifting of locally initial objects and universal (co)acting Hopf algebras

The universal (co) acting bi/Hopf algebras introduced by Yu. \, I. ~Manin, M. ~Sweedler and D. ~Tambara, the universal Hopf algebra of a given (co) module structure, as well as the universal group of a grading, introduced by J. ~Patera and H. ~Zassenhaus, find their applications in the classification of quantum symmetries. Typically, universal (co) acting objects are defined as initial or terminal in the corresponding categories and, as such, they do not always exist. In order to ensure their existence, we introduce the support of a given object, which generalizes the support of a grading and is used to restrict the class of objects under consideration. The existence problems for universal objects are formulated and studied in a purely categorical manner by seeing them as particular cases of the lifting problem for a locally initial object. We prove the existence of a lifting and, consequently, of the universal (co) acting objects under some assumptions on the base (braided or symmetric monoidal) category. In contrast to existing constructions, our approach is self-dual in the sense that we can use the same proof to obtain the existence of universal actions and coactions. In particular, when the base category is the category of vector spaces over a field, the category of sets or their duals, we recover known existence results for the aforementioned universal objects. The proposed approach allows us to apply our results not only to the classical categories of sets and vectors spaces and their duals but also to (co) modules over bi/Hopf algebras, differential graded vector spaces, G-sets and graded sets.