The Shape of Information: Entropy as a Topological Invariant

We present a unified categorical framework for information and entropy using the theory of (∞, n)-topoi constructed from a zero object. This approach demonstrates that Shannon entropy, von Neumann entropy, and thermodynamic entropy emerge as different manifestations of the same underlying topological structure. Starting from the vacuum state (zero object), we systematically build through free completions, ad- ditive closure, simplicial objects, and localization to obtain an (∞, 1)-topos T . The geometric realization H = Sh∞(CartSp) adds the differential structure necessary for physical applications. We show that entropy universally arises as a topological invari- ant of the cohesive structure, with the logarithm acting on connected components for information-theoretic entropies and dimension of de Rham cohomology for thermody- namic entropy. We then apply this framework to resolve fundamental open problems in quantum information theory, including additivity conjectures, distributed quantum coding, and entanglement-assisted communication limits, demonstrating that these challenges naturally require higher categorical structures.