What does this research mean for the field?

A new graphical language for closed symmetric monoidal categories enhances the expressiveness of internal hom functors using bracket wires. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To develop a graphical language that incorporates internal homs in closed symmetric monoidal categories.

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February 20, 2026Open Access

String Diagrams for Closed Symmetric Monoidal Categories

Key Points

To develop a graphical language that incorporates internal homs in closed symmetric monoidal categories.
Introduced string diagrams with special bracket wires for internal homs.
Established rules for interaction between morphism wires and bracket wires.
Demonstrated soundness and completeness of the diagrammatic calculus.
Developed a well-defined set of graphical rules for the new language.
Illustrated expressiveness with examples from category theory, logic, and programming semantics.
Showed how the new diagrammatic elements enhance understanding of internal hom functors.

Abstract

We introduce a graphical language for closed symmetric monoidal categories based on an extension of string diagrams with special bracket wires representing internal homs. These bracket wires make the structure of the internal hom functor explicit, allowing standard morphism wires to interact with them through a well-defined set of graphical rules. We establish the soundness and completeness of the diagrammatic calculus, and illustrate its expressiveness through examples drawn from category theory, logic and programming language semantics.

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