Linearly Distributive Fox Theorem

Linearly distributive categories (LDC), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivities. A seminal result in monoidal category theory is the Fox theorem, which characterizes cartesian categories as symmetric monoidal categories whose objects are equipped with canonical comonoid structures. The aim of this work is to extend the Fox theorem to LDCs and characterize the subclass of cartesian linearly distributive categories (CLDC). To do so, we introduce medial linearly distributive categories (MLDC), medial linear functors, and medial linear transformations. The former are LDCs which respect the logical medial rule, appearing frequently in deep inference, or alternatively are the appropriate structure at the intersection of LDCs and duoidal categories.

💡 Research Summary

The paper “Linearly Distributive Fox Theorem” extends the classical Fox theorem—originally characterising cartesian categories as symmetric monoidal categories whose objects carry canonical comonoid structures—to the setting of linearly distributive categories (LDCs). The author begins by recalling the Fox theorem in its traditional form, emphasizing that the right adjoint to the inclusion of cartesian categories into symmetric monoidal categories is obtained by taking the category of cocommutative comonoids. This adjunction underlies the identification of cartesian categories with those symmetric monoidal categories that possess a canonical comonoid on every object.

LDCs, introduced by Cockett and Seely, are categories equipped with two monoidal structures: a tensor (⊗, ⊤) and a par (⊕, ⊥), together with linear distributivity natural transformations δL and δR linking the two. While LDCs provide a natural categorical semantics for multiplicative linear logic, a direct analogue of the Fox theorem fails for arbitrary LDCs because the necessary interchange law between ⊗ and ⊕ is missing.

To overcome this obstacle the paper introduces the logical “medial” rule, a transformation of the form
 μ_{A,B,C,D} : (A⊗B)⊕(C⊗D) → (A⊕C)⊗(B⊕D).
This rule appears in deep inference systems and, categorically, coincides with the interchange maps of duoidal categories (categories with two monoidal structures and a set of compatibility maps). By enriching an LDC with the medial map together with additional structural morphisms
 Δ⊥ : ⊥ → ⊥⊗⊥, ∇⊤ : ⊤⊕⊤ → ⊤, m : ⊥ → ⊤,
the author defines medial linearly distributive categories (MLDCs). An MLDC can be viewed simultaneously as an LDC and a duoidal category; the medial map supplies the missing interchange law, while the extra maps make the unit objects behave like comonoid and monoid units respectively.

Within an MLDC the paper defines medial bimonoids (also called medial bimonoids). A medial bimonoid is an object A equipped with:

• a tensor‑diagonal Δ_A : A → A⊗A and tensor‑counit e_A : A → ⊤,
• a par‑multiplication ∇_A : A⊕A → A and par‑unit u_A : ⊥ → A, such that these four structure maps are compatible with the medial interchange μ. In other words, the usual bimonoid axioms hold, but the mixed compatibility is expressed precisely by the medial rule.

The central class of interest is cartesian linearly distributive categories (CLDCs), i.e. LDCs whose tensor structure is cartesian (so every object has a canonical ⊗‑diagonal and ⊗‑counit) and whose par structure is cocartesian (so every object has a canonical ⊕‑multiplication and ⊕‑unit). The paper proves that a CLDC is exactly a symmetric MLDC whose objects carry a coherent medial bimonoid structure. This is the analogue of the Fox theorem: the inclusion functor
 i : CLDC → SymMLDC
has a right adjoint R that sends a symmetric MLDC to the category of its medial bimonoids (which, by construction, form a CLDC). The adjunction is established at the level of 2‑categories, respecting both functors and natural transformations.

The proof proceeds by:

1. Showing that in any symmetric MLDC, the collection of medial bimonoids forms a symmetric monoidal category with tensor given by the cartesian product (thanks to the medial interchange) and par given by the cocartesian coproduct.
2. Verifying that this category satisfies the defining axioms of a CLDC (tensor cartesian, par cocartesian, and the required coherence diagrams).
3. Demonstrating that any strong symmetric monoidal functor between symmetric MLDCs preserves medial bimonoids, yielding a functor between the corresponding CLDCs, and that monoidal natural transformations are likewise preserved.
   Thus the right adjoint R is well‑defined on objects, 1‑cells, and 2‑cells, establishing the desired adjunction.

The paper also supplies a rich collection of examples: the category of sets with product and disjoint union, the category of relations, various bicategorical analogues (cartesian linear bicategories), and models arising from quantum computation where the tensor is the usual Hilbert‑space tensor product and the par is the direct sum, all equipped with appropriate medial maps. These examples illustrate how the medial rule naturally appears in concrete settings and how the theory unifies them under a single categorical framework.

In the concluding sections the author discusses further research directions: extending the medial rule to other logical systems (e.g., non‑commutative or substructural logics), exploring deeper connections between duoidal and linearly distributive structures, and applying the linearly distributive Fox theorem to the design of type systems for programming languages that combine linear and cartesian features (such as resource‑aware functional languages or quantum programming languages).

Overall, the paper delivers a substantial generalisation of the Fox theorem, bridging linear logic, deep inference, and duoidal category theory. By introducing MLDCs and medial bimonoids, it provides a clean characterisation of cartesian linearly distributive categories and opens a pathway for further categorical investigations of mixed monoidal structures.