[Paper Review] Graphical Presentations of Symmetric Monoidal Closed Theories

Richard Garner, Tom Hirschowitz+1 more|ArXiv.org|Oct 24, 2008

Logic, programming, and type systems14 references3 citations

TL;DR

This paper introduces a graphical calculus for symmetric monoidal closed (smc) theories using proof nets derived from Hughes' (2005) framework, extending it to handle equational theories via correctness criteria and rewiring equivalence. The key contribution is an isomorphism between a category of graphical nets and the free smc category generated by a signature, providing a fully graphical presentation of smc theories with equations.

ABSTRACT

We define a notion of symmetric monoidal closed (SMC) theory, consisting of a SMC signature augmented with equations, and describe the classifying categories of such theories in terms of proof nets.

Motivation & Objective

• To define symmetric monoidal closed (smc) theories as signatures augmented with equations, generalizing algebraic theories to monoidal closed structure.
• To provide a graphical, proof net-based presentation of the free smc category generated by such a theory, enabling visual and combinatorial reasoning.
• To extend Hughes' (2005) proof net framework—originally for free smc categories on a set of sorts—to handle equational theories via correctness and rewiring.
• To establish a categorical isomorphism between a graphical category of nets and the classifying category of an smc theory, ensuring full faithfulness and full soundness.

Proposed method

• Define an smc signature using a set of sorts $X$ and a graph of terms in the free smc category over $X$, with operations including $\otimes$, $\multimap$, $I$, and structural isomorphisms.
• Construct derived terms via closure under composition, identities, and structural isomorphisms such as $\alpha_{abc}$, $\lambda_a$, $\rho_a$, $\sigma_{ab}$, $\epsilon_{ab}$, and $\eta_{ab}$.
• Represent morphisms as prenets: a support $C$ and a directed graph $G$ with a partial incidence function $g$ that is bijective on ports of each sort $x \in X$, modulo isomorphism of supports.
• Define correctness of a prenet via the condition that all switchings of its associated classical MLL formulae are trees, generalizing Danos and Regnier’s criterion.
• Introduce rewiring equivalence: two nets are equivalent if one can be transformed into the other by changing the target of a single edge incident to a negative occurrence of $I$, preserving correctness.
• Prove that the quotient category $\mathcal{D}_{X,\Sigma}$, formed by identifying rewired nets, is isomorphic to the free smc category $\mathcal{C}_{X,\Sigma}$, establishing the graphical calculus as a sound and complete presentation.

Experimental results

Research questions

• RQ1 How can symmetric monoidal closed theories be formally defined as signatures with equations, extending the notion of algebraic theories to monoidal closed structure?
• RQ2 Can Hughes’ proof net framework be extended to model free smc categories generated by equational theories, not just free ones?
• RQ3 What graphical correctness criterion ensures that a net represents a valid morphism in the free smc category?
• RQ4 How can equivalence of graphical representations be captured while preserving categorical structure, and what is the role of rewiring in this?
• RQ5 Is the category of graphical nets, modulo rewiring, isomorphic to the classifying category of the smc theory?

Key findings

• The category $\mathcal{D}_{X,\Sigma}$ of graphical nets modulo rewiring is isomorphic to the free symmetric monoidal closed category $\mathcal{C}_{X,\Sigma}$ generated by a given smc signature $(X,\Sigma)$, establishing a full graphical calculus.
• Correctness of a net is defined via the tree condition on all switchings of the associated classical MLL formulae, generalizing Danos and Regnier’s criterion to the equational setting.
• Rewiring equivalence preserves correctness and induces a well-defined equivalence relation on nets, allowing for a quotient category that captures the same structure as the syntactic free smc category.
• The construction provides a faithful and full graphical representation of morphisms in the free smc category, with composition defined by graph gluing along shared ports.
• The framework extends Hughes’ (2005) proof net presentation to handle equational theories by reducing the general case to the free case via support-based nets.
• The result generalizes Trimble’s rewiring and Kelly-MacLane graph constructions, embedding them into a unified framework for smc theories with equations.