Abstract
We introduce a hierarchical categorical framework for Schema-Based Learning (SBL) structured across four interconnected levels. At the schema level, a free multicategory Sch_{syn} encodes fundamental schemas and transformations. An implementation functor \mathcal{I} maps syntactic schemas to representational languages, inducing via the Grothendieck construction the total category Sch_{impl}. Implemented schemas are mapped by a functor Model into the Kleisli category \mathbf{KL(G)} of the Giry monad, yielding probabilistic models, while an instances presheaf assigns evaluated instance spaces. A semantic category Sch_{sem}, defined as a full subcategory of \mathbf{KL(G)}, provides semantic grounding through an interpretation functor from Sch_{impl}.
At the agent level, Sch_{impl} is equipped with a duoidal structure \mathcal{O}_{Sch} supporting schema-based workflows. A left duoidal action on the category Mind enables workflow execution over mental objects, whose components include mental spaces, predictive models, and a cognitive kernel composed of memory and cognitive modules. Each module is specified by schema-typed interfaces, duoidal workflows, a success condition, and a logical signature. Memory is formalized categorically via memory subsystems, a presheaf Data_M, a monoidal operation category Ops_M, and read/write natural transformations. Together with the Body category, Mind defines the embodied SBL agent.
At higher levels, SBL is represented as an object of the agent architecture category ArchCat, enabling comparison with heterogeneous paradigms, while the World category models multi-agent and agent-environment interactions. Altogether, the framework forms a weak hierarchical n-categorical structure linking schema semantics, cognition, embodiment, architectural abstraction, and world-level interaction.
