Free Doubly-Infinitary Distributive Categories are Cartesian Closed
This paper investigates *bifinitely distributive categories*—categories in which products distribute over coproducts and coproducts distribute over products—in the infinitary setting, clarifying their logical relationships with universality, infinitary distributivity, and Cartesian closure. Method: Employing category theory, infinitary limit/colimit theory, universal algebra, and adjoint functor techniques, the notion is formally axiomatized for the first time. Contribution/Results: We rigorously prove that every free bifinitely distributive category is necessarily Cartesian closed. Several nontrivial concrete examples are constructed to enable comparative analysis and validate the expressiveness of the framework—demonstrating it strictly subsumes classical infinitary distributive categories. The work also identifies open problems, including the existence of non-canonical isomorphisms. These results establish a novel categorical foundation for higher-order type theory and logical semantics.