Kleisli Semantics and Hypergraph Composition for Greimasian Narrative Programs

This article proposes a category-theoretic formalization of Greimasian narrative programs (NPs) that makes their compositional structure mathematically precise. Building on a reconstruction of the actantial model as a categorical schema, we introduce a refined typological schema of actants and derive Set-valued instances corresponding to role-indexed elements of a narrative. NPs are represented within a categorical schema whose selected morphisms are interpreted using monadic semantics on Set. In particular, the List monad provides Kleisli semantics for modeling non-atomic, list-valued actantial configurations, while the Maybe monad encodes optional dependencies between programs. This yields a compact, structured representation of narrative programs as structured data with an intrinsic compositional interpretation. To account for the compositional dynamics of narrative formation, we lift these constructions into a diagrammatic setting by freely generating a symmetric monoidal category from the set of actants, adjoining narrative program generators, and subsequently equipping the resulting structure with Frobenius operations to obtain a hypergraph category. In this framework, narrative programs act as generators of morphisms, and their composition is realized through wiring diagrams. A narrative trajectory is thereby realized categorically as a single composite morphism. This approach provides a unified mathematical framework for structural semiotics, connecting data-level representations of narrative elements with their compositional realization in discourse.