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

TL;DR

This paper establishes a category-theoretic independence theorem for meaning, object, name, and existence within topos theory, demonstrating their distinct structural levels and non-recoverability from one another.

Contribution

It introduces a formal separation of fundamental notions in categorical semantics, especially highlighting the distinction between internal existence and global naming.

Findings

Objects can be internally inhabited without global elements.

Meaning, object, name, and existence occupy separate structural levels.

The results underpin the use of geometric universes as foundational frameworks.

Abstract

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.