A Topos-Theoretic Framework for Artistic-Oriented Knowledge Graphs

This paper introduces a two-level framework for Artistic-Oriented Knowledge Graphs (AOKGs). The lower level is a knowledge graph capturing factual entities (artworks, artists, symbols, techniques) and their relationships. The upper level is a topos—a category of sheaves—constructed over the knowledge graph, which formalizes what Alain Badiou calls the 'logic of appearing.' The topos provides an internal logic (generally intuitionistic) that governs how meanings are assigned to entities in context-dependent ways. Sheaves assign to each region of the knowledge graph a set of possible interpretations (e.g., iconographic meanings, stylistic categories, aesthetic values). The sheaf condition ensures local consistency and global coherence. An artistic 'event' is formalized as a change of topos: when a new artwork cannot be interpreted within the existing topos, a new topos with a different logic must be constructed, leading to a truth procedure that reinterprets prior works and generates new meanings. The paper illustrates this with examples: the emergence of non-objective art (Malevich's Black Square) and Cubism (Picasso's Les Demoiselles d'Avignon). The framework also provides formal definitions of beauty (perfect exemplar of current topos) and the sublime (gesturing beyond current topos). Implementation steps include building the knowledge graph, defining a site with a Grothendieck topology, constructing sheaves manually or via machine learning, and computing the internal logic. Applications include multi-perspective interpretation, event detection, truth procedure tracing, and aesthetic reasoning. The framework is theoretical and conceptual, with no quantitative results, but it offers a rigorous mathematical foundation for computational art history and digital aesthetics.