Working Paper: Towards Schema-based Learning from a Category-Theoretic Perspective

arXiv:2604.10589v1 Announce Type: new
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 $mathcalI$ 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 $mathbfKL(G)$ 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 $mathbfKL(G)$, provides semantic grounding through an interpretation functor from $Sch_impl$.
At the agent level, $Sch_impl$ is equipped with a duoidal structure $mathcalO_Sch$ 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.