Topos-Semantic Foundations for Agentic Understanding: Relevance, Judgment, Commitment, and Curiosity-Driven Agency

Topos-Semantic Foundations for Agentic Understanding: Relevance, Judgment, Commitment, and Curiosity-Driven Agency develops a topos-theoretic semantics for “agentic” cognition in which context, compression, learning, and curiosity-driven action arise as structural constructions rather than heuristic add-ons. Context is modeled as a relevance slice topos (semantic localization relative to a focus c), with relevance transport given canonically by pullback along focus morphisms. Candidate explanations are represented as interpreted finite graph-like shapes (dessin/ribbon-graph style), whose semantic content is obtained by colimits (integrative understanding), while judgment is enforced via limits against evidence and constraints (compatibility / constraint satisfaction). Conceptual commitment is modeled as sheafification with respect to focus-indexed Lawvere–Tierney modalities, yielding a principled notion of compression as stability under admissibility. Learning is treated as conservative accommodation: a minimal enlargement of modality j to j' required to admit new judged content, penalized by a probe-based accommodation cost reflecting semantic disturbance. 

The framework assembles into a fibrational semantics over relevance (via Grothendieck construction), interpreting agent states as sections and learning episodes as coherence-preserving 2-cells; under suitable Beck-Chevalley-style coherence, Explain/Judge/Commit commute with relevance transport. To connect understanding with action, the paper introduces an action fibration and defines value of information internally as expected reduction in a judgment-defect functional, producing a curiosity-driven decision principle that balances epistemic gain against accommodation cost.

Philosophical and pedagogical influences for the theory are grounded by the work and concepts developed by Xavier Zubiri, Bernard Lonergan, Lev Vygotsky, and Jean Piaget. In particular, Zubiri and Lonergan motivate the triad understanding → judgment → commitment, while Vygotsky and Piaget inform the learning picture as scaffolded expansion of admissibility (minimal modality enlargement) rather than unconstrained model revision.

Overall, the theory is designed to be compositional under inquiry change, conservative in learning, and suitable as a stable semantic substrate for executable agent architectures (with realizability-topos instantiations highlighted as a next step). In other words, the topos-semantic model is meant to replace “tokens-first” thinking with a meaning-first semantics. Instead of treating context as a sliding window of tokens processed by attention, it treats context as semantic localization: a change of focus is essentially a pullback along a focus morphism in a relevance slice.

This will be known as Paper I going forward.