The Private Topos of the Ineffable : From Fibers of Meta-Ethical Dilemmas to Hypercohomology

This study aims to formalize meta-ethical dilemmas arising between the internal generation of meaning within the self ⟨I⟩ and linguistic practices and norm formation within a community, by integrating probabilistic sheaf theory, category theory, and topos theory. From the perspective of Wittgensteinian language games, meaning does not exist as a pre-given, fixed entity; rather, it appears as an ineffable dilemma—a process of local selection and revision that emerges only through context-dependent use. In this work, such local meaning selection associated with this ineffable dilemma is modeled as probabilistically valued sheaf data distributed over a base space. Contextual overlaps and inconsistencies among these data are described by a Total complex that integrates a sheaf complex with a Čech complex. This Total complex provides a minimal computational framework capable of simultaneously capturing both the local coherence of meaning practices and the transition structures between distinct contexts. However, the framework of sheaf cohomology alone is insufficient to adequately describe situations in which meaning practices are locally coherent yet globally non-integrable—namely, the preservation structure of dilemmas. To address this limitation, we introduce hypercohomology associated with the Total complex and theoretically position dilemmas not as contradictions to be resolved, but as topological obstructions inherent in contextual overlaps. Furthermore, this paper formalizes the world of the self ⟨I⟩ and that of the community as distinct topoi, each equipped with its own internal logic and rules of meaning generation. These topoi are not reducible to a single logical space; rather, they are related through morphisms that fail to form a complete adjunction, giving rise instead to a semi-adjoint structure. Crucially, this adjunction failure—that is, the principled impossibility of fully translating or identifying meaning, norms, and judgments—constitutes the ontological origin of dilemmas themselves. Within this topos-theoretic framework, dilemmas are not contingent problems arising from epistemic insufficiency or lack of information on the part of the self, but are preserved as necessary structural consequences of the coexistence of multiple meaning-worlds. Accordingly, ethical judgment and the subjectivity of the self ⟨I⟩ are not destined to converge toward a single universal norm; rather, they are dynamically constituted through the relations between topoi, their morphisms, and the ruptures thereof. In conclusion, this study does not seek to subsume meaning, ethics, and subjectivity into a unified system, but instead presents a new theoretical foundation that describes them as relational structures across Total complexes and multiple topoi, thereby theoretically demonstrating potential applications to AI ethics and semantic learning models.