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

Abstract

We introduce a hierarchical categorical framework for Schema-Based Learning (SBL) structured across four interconnected levels. At the schema level, a free multicategory $Sc{h}_{syn}$ encodes fundamental schemas and transformations. An implementation functor $\mathcal{I}$ maps syntactic schemas to representational languages, inducing via the Grothendieck construction the total category $Sc{h}_{impl}$. Implemented schemas are mapped by a functor $Model$ into the Kleisli category $\mathbf{KL}(\mathbf{G})$ of the Giry monad, yielding probabilistic models, while an instances presheaf assigns evaluated instance spaces. A semantic category $Sc{h}_{sem}$, defined as a full subcategory of $\mathbf{KL}(\mathbf{G})$, provides semantic grounding through an interpretation functor from $Sc{h}_{impl}$. At the agent level, $Sc{h}_{impl}$ 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 $Dat{a}_M$, a monoidal operation category $Op{s}_M$, 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.

43 pages, 3 figures