Grothendieck's Geometric Universes and A Sheaf-Theoretic Foundation of Information Network
Takao Inoué
TL;DR
The paper proposes Grothendieck's geometric universes as a sheaf-theoretic foundation for information networks, where distributed information is organized via topoi and internal logic emerges from geometric structure. By modeling local informational states as sections of a sheaf on a site $(\mathcal{C},J)$ and using the internal language of a topos, logical validity is shown to be an intrinsic consequence of the framework. The author introduces intrinsic logicism as a contemporary, internalized version of the Frege-Russell program, realized through the internal semantics of geometric universes rather than external axioms. This approach offers a principled foundation for knowledge representation and consensus in distributed, context-aware systems, with potential practical applications in information networks and related domains.
Abstract
This paper proposes an interpretation of Grothendieck's geometric universes as a foundational framework for \emph{information networks}. We argue that Grothendieck topologies, sheaves, and topoi provide a sheaf-theoretic semantics in which distributed and locally held information can be integrated into globally coherent structures. In this setting, local informational states are represented by sections, while the sheaf condition governs consistency, agreement, and consensus across a network. Logical validity and mathematical existence are therefore not imposed externally but arise intrinsically from geometric and categorical conditions. From this perspective, Grothendieck's geometric universes constitute a natural foundation for information networks governed by intrinsic logical principles. Moreover, we propose that Grothendieck's geometric universes themselves concretely instantiate what the author calls \emph{intrinsic logicism}. This position is intended as a contemporary reconstruction of the classical logicist program of Frege and Russell, reformulated within the framework of category theory and topos theory, where logical structure is generated internally by geometric and categorical organization rather than presupposed as an external foundational layer.