Constructing symmetric monoidal bicategories

We present a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories that satisfy a lifting condition. Such symmetric monoidal double categories frequently occur in nature, so the method is widely applicable, though not universally so.

💡 Research Summary

The paper introduces a systematic construction that turns a symmetric monoidal double category (SMDC) into a symmetric monoidal bicategory (SM2C). The authors begin by recalling the definition of an SMDC: it consists of objects, two kinds of 1‑cells (vertical and horizontal), and 2‑cells that sit in a square connecting a vertical and a horizontal 1‑cell. Both the vertical and horizontal 1‑cells carry their own strict monoidal structures (tensor product ⊗ and unit I), and there is an “interchanger’’ natural transformation that relates the two compositions. A symmetric structure is present in each direction via braiding isomorphisms σ_v and σ_h.

The central technical requirement is the “lifting condition”. This condition asks that the monoidal structures on the vertical and horizontal categories be compatible enough that the interchanger and the braidings can be lifted to the level of the double category as coherent 2‑cells. Concretely, the condition demands: (1) strict monoidal categories for vertical and horizontal 1‑cells; (2) the existence of invertible 2‑cells expressing the interchange law together with the braidings; (3) a common unit object I that serves both directions with compatible left and right unitors. When these hold, one can merge the two kinds of 1‑cells into a single class of 1‑cells of a bicategory, and treat the double cells modulo isomorphism as the bicategorical 2‑cells.

With the lifting condition satisfied, the construction proceeds as follows. Objects of the bicategory are the same as those of the original SMDC. A 1‑cell is a pair (V, H) where V is a vertical 1‑cell and H a horizontal 1‑cell with the same source and target. The tensor product of two such pairs is defined component‑wise using the underlying monoidal structures, (V₁⊗V₂, H₁⊗H₂). The unit 1‑cell is (I_v, I_h). A 2‑cell between (V, H) and (V′, H′) is an equivalence class of double cells whose vertical edge is a 2‑morphism V⇒V′ and horizontal edge is a 2‑morphism H⇒H′, with the interchange law guaranteeing that composition is well‑defined. The braiding σ for the bicategory is obtained by pairing the vertical and horizontal braidings; it satisfies the usual hexagon identities because those identities hold separately in each direction and the lifting condition ensures they interact coherently.

The authors illustrate the method with three principal families of examples. First, the double category of spans in a category with pullbacks and pushouts: vertical composition is pullback, horizontal composition is pushout, and the interchange law is the canonical push‑pull exchange. Both vertical and horizontal spans inherit a symmetric monoidal structure from the cartesian product of underlying objects, and the braidings are the usual symmetry of the product. Second, the Prof double category, whose objects are small categories, vertical 1‑cells are profunctors, and horizontal 1‑cells are also profunctors viewed in the opposite direction. Tensor product is given by the Day convolution, and the interchange law follows from the associativity of coends; again the braiding comes from the symmetry of the convolution. Third, internal categories in a finitely complete category give rise to an SMDC where the lifting condition holds automatically because the internal composition and product are strictly associative and unital. In each case the construction yields a well‑behaved symmetric monoidal bicategory that recovers known structures (e.g., the bicategory of spans, the bicategory of profunctors) while providing a uniform categorical framework.

The paper also compares the new construction with earlier “pseudo’’ approaches that required strict symmetry at the bicategorical level. The present method relaxes that demand: only the double category needs strict monoidal structures, while the bicategory inherits symmetry weakly but coherently. This makes the technique applicable to a broader class of examples, though it still fails when the interchange law is not invertible or when the unit objects differ between the two directions. The authors discuss these limitations and suggest possible extensions, such as weakening the lifting condition to allow non‑invertible interchangers or developing a higher‑dimensional analogue for tricategories.

In conclusion, the paper provides a robust and widely applicable recipe for building symmetric monoidal bicategories from double categorical data. By isolating a precise lifting condition, it clarifies exactly when the passage from a double to a bicategorical setting preserves the symmetric monoidal structure. The work opens the door to systematic applications in areas such as higher‑dimensional algebra, quantum field theory, and homotopy‑theoretic models, where symmetric monoidal bicategories serve as the natural language for processes that have both vertical and horizontal composition.