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Kleisli semantics and hypergraph composition for Greimasian narrative programs

Michael Fowler

Abstract

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 morphisms are interpreted using monads on Set. In particular, the List monad provides a Kleisli semantics for modeling non-atomic, list-valued actantial configurations, while the Maybe monad encodes optional dependencies between programs. This yields a minimal representation of narrative programs as structured data with an intrinsic compositional interpretation. To account for the dynamics of narrative formation, we lift these constructions into a diagrammatic setting by freely generating a symmetric monoidal category, and subsequently a hypergraph category, from the set of actants. In this framework, narrative programs act as generators of morphisms, and their composition is realized through wiring diagrams. A narrative trajectory is thereby interpreted 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.

Kleisli semantics and hypergraph composition for Greimasian narrative programs

Abstract

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 morphisms are interpreted using monads on Set. In particular, the List monad provides a Kleisli semantics for modeling non-atomic, list-valued actantial configurations, while the Maybe monad encodes optional dependencies between programs. This yields a minimal representation of narrative programs as structured data with an intrinsic compositional interpretation. To account for the dynamics of narrative formation, we lift these constructions into a diagrammatic setting by freely generating a symmetric monoidal category, and subsequently a hypergraph category, from the set of actants. In this framework, narrative programs act as generators of morphisms, and their composition is realized through wiring diagrams. A narrative trajectory is thereby interpreted 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.

Paper Structure

This paper contains 10 sections, 2 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Summary
  3. Formalising narrative programs
  4. Categorical schemas and the actantial model
  5. Kleisli database instances
  6. Diagramming discoursivization
  7. Narrative trajectories in a hypergraph category
  8. Discussion
  9. Conclusion
  10. Appendix

Key Result

Proposition 1

Let $I: \mathcal{A}^{\prime}\rightarrow \mathbf{Set}$ be the actantial instance with underlying set $X$. Then there exists a free symmetric monoidal category generated by $\mathbf{Disc}(X)$. Its objects are finite tensor products of elements of $X$, including the unit object $I$, and its morphisms are generated by identities, symmetries, and composition. Moreover, for any symmetric monoidal categ

Figures (4)

  • Figure 1: (a) Categorical schema $\mathcal{A}$ of the Greimas actantial model with path equivalences: $a_{3}\circ a_{4}\simeq a_{6}$, and $a_{3}\circ a_{5}\simeq a_{7}$. (b) Ontological log of the categorical schema $\mathcal{A}$ of the Greimas actantial model with equivalent aspects given as: $(\textrm{seeks to cojoin with})\circ (\textrm{assists})\simeq (\textrm{assists a conjunction with})$, and $(\textrm{seeks to cojoin with})\circ (\textrm{hinders})\simeq (\textrm{hinders a conjunction with})$.
  • Figure 2: Database table of instances of the object $P \in \mathrm{Ob}(\mathcal{N})$ and olog of the categorical schema $\mathcal{N}$.
  • Figure 3: Wiring diagrams of the narrative programs NP4 and NP5 (cf. Table \ref{['TableN']}) from The Hare & the Tortoise.
  • Figure 4: The wiring diagram $\mathcal{W}_{0}(\cap_{\textrm{NP1}},\cap_{\textrm{NP5}},\cap_{\textrm{NP4}},\cap_{\textrm{NP3}},\cap_{\textrm{NP6}};\mathsf{NT})$ of the narrative trajectory of the Aesop fable The Hare & Tortoise.

Theorems & Definitions (29)

  • Definition 1: Categorical schema
  • Definition 2: Instance on a schema
  • Definition 3: Refined typological schema of actants
  • Remark 1
  • Definition 4: Actantial instance
  • Definition 5: Monad
  • Definition 6: Kleisli category
  • Definition 7: Kleisli $\top$-instance
  • Definition 8: List-valued narrative program
  • Remark 2
  • ...and 19 more