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 $\mathbf{Sh}(S^1)$-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.

翻译：我们证明了关于四个基本概念——意义、对象、名称与存在——的一个范畴论独立性定理。在劳维尔风格的范畴语义学中，特别是在拓扑斯中工作，我们表明这些概念占据不同的结构层次（对象、态射、元素与内部逻辑层次），且无法从彼此中一致地恢复。关键的分离出现在内部存在与全局命名之间。利用拓扑斯 $\mathbf{Sh}(S^1)$——一个非平凡覆盖的局部截面层——中的一个具体例子，我们展示了一个对象，它在内部被居住但不允许任何全局元素。这些结果为将几何宇宙视为信息网络的基础框架提供了精确的结构基础。