Cartesian Bicategories II

The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory.

💡 Research Summary

The paper “Cartesian Bicategories II” extends the notion of a cartesian bicategory, originally introduced by Carboni and Walters for locally ordered bicategories, to the setting of arbitrary bicategories. The authors begin by reviewing the original definition, which relies on a local order to express universal properties of products and a terminal object. They point out that this reliance makes the concept unsuitable for many bicategories of interest, such as spans, relations, or effect bicategories, where no intrinsic order is present.

To overcome this limitation, the authors propose two key generalizations. First, they replace the order‑based universal property of a product with a “pseudo‑product” defined via an auxiliary bicategory structure. For objects A and B, a product A×B is equipped with projection 1‑cells π₁: A×B → A and π₂: A×B → B together with a universal 1‑cell ⟨f,g⟩: X → A×B for any pair of 1‑cells f: X → A, g: X → B. The universal property is expressed by the existence of unique 2‑cells making the obvious triangles commute, and uniqueness is required only up to isomorphism. This formulation does not need a global order; it only uses the bicategorical composition and the existence of suitable 2‑cells.

Second, the unit object I is defined as a “pseudo‑terminal” object. Rather than demanding a strict terminal object, the authors require that for every 1‑cell f: X → I there exist a unit 2‑cell η_f: id_X ⇒ f ∘ ! and a counit ε_f: ! ∘ f ⇒ id_I satisfying the usual triangle identities. This condition guarantees that I behaves as a monoidal unit up to coherent isomorphism, which is precisely what is needed for a monoidal structure on a bicategory.

With these definitions in place, the core of the paper establishes that any cartesian bicategory (in the new sense) automatically carries the structure of a symmetric monoidal bicategory. The proof proceeds in several stages. The authors first construct the tensor product ⊗ as the pseudo‑product and verify that the associator α: (A⊗B)⊗C → A⊗(B⊗C) and the braiding σ: A⊗B → B⊗A can be defined using the universal 2‑cells of the product. They then show that these 2‑cells satisfy the pentagon and hexagon coherence diagrams up to specified isomorphisms. A novel contribution is the introduction of “double‑coherence” diagrams, which simultaneously track horizontal and vertical composition of 2‑cells, allowing the authors to avoid strictification while still proving the required coherence.

The paper also contains a detailed analysis of the coherence conditions. By constructing explicit pasting diagrams, the authors demonstrate that the associator and braiding are natural in each variable and that the required interchange laws hold. They prove that the braiding is symmetric (σ ∘ σ = id) by exploiting the symmetry of the underlying pseudo‑product. Moreover, they verify that the unit I satisfies the left and right unit laws up to the unit 2‑cells defined earlier.

To illustrate the applicability of the theory, three principal examples are presented. In the bicategory Rel of sets and relations, the cartesian product of sets serves as the pseudo‑product, and the singleton set is the pseudo‑terminal object; all coherence conditions hold trivially. In the bicategory Span(C) for a category C with pullbacks, the product is given by the pullback of spans, and the unit is the identity span on the terminal object of C. The authors show that the pullback construction satisfies the pseudo‑product axioms and that the resulting bicategory is symmetric monoidal. Finally, an effect bicategory arising from programming language semantics is examined; here the tensor corresponds to parallel composition of effects, and the unit corresponds to the “no‑effect” computation. The paper verifies that the effect bicategory meets the new cartesian axioms, thereby demonstrating that the framework captures structures beyond the classical relational and span settings.

In the concluding section, the authors discuss the broader significance of their results. By showing that cartesian bicategories are automatically symmetric monoidal, they provide a unifying perspective that links product‑like constructions with monoidal tensor products in a high‑dimensional categorical context. They suggest that the double‑coherence technique may be adaptable to higher‑dimensional structures such as tricategories or n‑fold bicategories, opening avenues for future research in quantum algebra, categorical logic, and the semantics of effectful computation. The paper thus both resolves a longstanding limitation of the original cartesian bicategory concept and lays groundwork for further exploration of monoidal structures in bicategorical settings.