Generalized crossed hypermodules and hypergroup-groupoids

In this paper, we construct a categorical framework for the study of generalized crossed hypermodules and their fundamental connection with generalized hypergroup-groupoids. This approach extends the classical theory of crossed modules to generalized hypergroups, where the binary operation may be partially defined or subject to generalized associativity conditions. We introduce the notion of a generalized hypergroup-groupoid, unifying groupoids and generalized hypergroups, and prove a categorical equivalence between the category of generalized crossed hypermodules and that of generalized hypergroup-groupoids with abelian object sets. Beyond its structural significance, this theory provides an algebraic framework for modeling non-classical symmetries and multi-valued interactions. We present applications in (1) algebraic topology, via generalized homotopy groupoids and higher cohomological actions; (2) quantum and statistical algebra, through multi-state symmetries governed by non-invertible operations; (3) automata and computation theory, by encoding nondeterministic transitions within hypergroupoid structures; (4) network and systems theory, using categorical models of multi-agent interaction spaces; and (5) fuzzy and soft algebraic systems, by formalizing uncertainty and graded associativity through generalized hyperoperations. These examples show that generalized crossed hypermodules provide a unifying language linking algebraic topology, categorical algebra, and non-classical structural models, and suggest further developments in higher-dimensional and homotopical algebra.