On the Category-Theoretic Independence of Meaning, Object, Name and Existence

We prove a category-theoretic independence theorem for four fundamental notions: meaning, object, name, and existence. Working in a Lawvere-style categorical semantics and in particular in toposes, we show that these notions occupy distinct structural levels (object, morphism, element, and internal logical level) and are not uniformly recoverable from one another. The key separation arises between internal existence and global naming. Using a concrete example in the topos $\mathbf{S}\mathbf{h}(S^1)$-the sheaf of local sections of a nontrivial covering-we exhibit an object that is internally inhabited but admits no global element. These results provide a precise structural basis for treating geometric universes as foundational frameworks for information networks.