---
title: Symmetric Monoidal Double Category
url: https://www.emergentmind.com/topics/symmetric-monoidal-double-category
type: topic
---

# Symmetric Monoidal Double Category

A symmetric monoidal double category is a higher-categorical structure that combines the features of both double categories and symmetric monoidal categories, providing an organizational framework that encodes two types of morphisms (vertical and horizontal), 2-dimensional "squares," and coherently compatible tensor product operations equipped with symmetric braiding. This construction has become a foundational object in modern higher category theory, with applications ranging from modular functors in conformal field theory to the categorical study of spans, profunctors, and multicategories. 

## 1. Structure of Double Categories and Monoidal Double Categories

A **double category** $\mathbb{D}$ consists of:
- A collection of objects $\mathrm{Ob}(\mathbb{D})$.
- For each pair $A,B\in\mathrm{Ob}(\mathbb{D})$, a set of vertical 1-morphisms $A\xrightarrow{v}B$ with strictly associative composition and identities.
- For each pair $A,B\in\mathrm{Ob}(\mathbb{D})$, a set of horizontal 1-cells $A\xRightarrow{f}B$, with composition associative and unital up to coherent isomorphism.
- 2-cells (squares) of the form:
  $$
  \begin{tikzcd}[ampersand replacement=\&]
    A \ar[r,"M"] \ar[d,"f"'] \ar[dr,phantom,"\Downarrow \alpha"] \& B \ar[d,"g"] \\
    C \ar[r,"N"'] \& D
  \end{tikzcd}
  $$
  with vertical sources and targets given by the vertical morphisms, and horizontal sources and targets given by the horizontal 1-cells. Both vertical and horizontal compositions of squares are defined, subject to the interchange law:
  $$
  (\alpha'\circ_h \alpha)\circ_v (\beta'\circ_h \beta) = (\alpha'\circ_v \beta')\circ_h (\alpha \circ_v \beta)
  $$
  ([1004.0993], [1910.09240], [2605.03708]).

A **monoidal double category** $(\mathbb{D},\otimes,I,\alpha,\lambda,\rho)$ is a double category equipped with:
- A (pseudo) double functor (the tensor product) $\otimes: \mathbb{D}\times\mathbb{D} \to \mathbb{D}$, which acts on all levels (objects, vertical, horizontal, and 2-cells).
- A unit object $I$ (provided as a double functor $\mathbbm{1}\to\mathbb{D}$).
- Associator and unitors: invertible double-natural transformations
  $$
  \alpha: \otimes(\otimes\times \mathrm{Id}) \Rightarrow \otimes(\mathrm{Id}\times \otimes),\quad
  \lambda: \otimes(I \times \mathrm{Id}) \cong \mathrm{Id},\quad
  \rho: \otimes(\mathrm{Id} \times I) \cong \mathrm{Id}
  $$
  These satisfy the pentagon and triangle coherence axioms ([1910.09240], [1004.0993]).

## 2. Symmetry, Braiding, and Coherence Conditions

A **braided monoidal double category** includes a braiding, given as an invertible double-natural transformation:
$$
\beta: \otimes \Rightarrow \otimes\circ\tau
$$
where $\tau$ swaps the factors in $\mathbb{D}\times\mathbb{D}$. The braiding satisfies the hexagon axiom (the corresponding diagram involving associators and $\beta$ commutes). 

A **symmetric monoidal double category** is a braided monoidal double category in which the double braiding squares to the identity:
$$
\beta_{B,A}\circ\beta_{A,B} = \mathrm{id}_{A\otimes B}
$$
These conditions extend naturally to all categorical levels (vertical, horizontal, and 2-cells) and are encoded in the required commutative diagrams ([1910.09240], [1004.0993], [2605.03708], [2511.14402]).

## 3. Fibrancy and the Lifting Condition

To transfer monoidal, braided, or symmetric structures from the double categorical setting to associated bicategories, a **fibrancy** (or "lifting") condition is required. Specifically, for every vertical 1-morphism $f: A\to B$, there must exist a **companion** horizontal 1-cell $\widehat{f}: A\hto B$, together with canonical 2-cells ("zig-zag" triangles) and their duals (conjoints), satisfying specified identities. The fibrancy condition ensures that the induced structure on the horizontal bicategory inherits the desired monoidal properties ([1004.0993], [1910.09240]).

## 4. Symmetric Monoidal Double Categories and Their Horizontal Bicategories

Given a fibrant symmetric monoidal double category $\mathbb{D}$, the **horizontal bicategory** $\mathcal{H}(\mathbb{D})$ is defined as follows:
- **Objects:** same as $\mathbb{D}_0$ (the objects of the underlying vertical category).
- **1-cells:** horizontal 1-cells of $\mathbb{D}$, $M: A\hto B$.
- **2-cells:** globular 2-cells, i.e., squares whose vertical edges are both identities.

The symmetric monoidal structure on $\mathbb{D}$ descends to $\mathcal{H}(\mathbb{D})$:
- The tensor product on objects and 1-cells comes from the corresponding structures in $\mathbb{D}$.
- Associators, unitors, and braidings are given by the companions of the corresponding vertical arrows in $\mathbb{D}$.
- All coherence diagrams (pentagon, triangle, hexagon, syllepsis) are induced via the uniqueness and compatibility properties of companions.

The association $\mathbb{D}\mapsto \mathcal{H}(\mathbb{D})$ extends to a functor between (fibrant) symmetric monoidal double categories and symmetric monoidal bicategories, preserving the full algebraic structure ([1004.0993], [1910.09240]).

## 5. Canonical Examples

Several prominent examples illustrate the ubiquity and utility of symmetric monoidal double categories:
- **Bordisms:** The double category $\mathbb{D}_{\mathrm{bord}}$ of 1-manifolds, smooth embeddings (vertical), 2-dimensional bordisms (horizontal), and isotopy classes of embeddings of bordisms (squares), with disjoint union as the symmetric monoidal structure ([2605.03708]).
- **Spans:** For any category $\mathbf{C}$ with pullbacks, the double category $\mathrm{Span}(\mathbf{C})$ takes objects from $\mathbf{C}$, horizontal 1-cells as spans, vertical arrows as morphisms in $\mathbf{C}$, and squares as maps of spans. It is symmetric monoidal under the cartesian product.
- **Profunctors:** The double category $\mathrm{Prof}$, with small categories as objects, functors as vertical arrows, profunctors as horizontal arrows, and natural transformations as 2-cells, is symmetric monoidal under the cartesian product ([1910.09240], [1004.0993]).
- **Open Markov Processes:** Combinatorially, one obtains further examples by considering decorated cospans or similar structures under disjoint union.
- **Multicategories and Operads:** The double category of symmetric multicategories, multifunctors, and bimodules inherits a symmetric oplax monoidal structure, subsuming constructions such as the Boardman-Vogt tensor product ([2511.14402]).

## 6. The Gray Monoidal Product and Closed Structure

The category of double categories and double functors $\mathrm{DblCat}$ admits a symmetric closed monoidal structure via the **Gray tensor product** $\otimes$:
- For double categories $A$ and $B$, $A\otimes B$ is constructed as the left adjoint to the internal hom $[A,-]$.
- The unit object $I$ is the terminal double category.
- There are canonical associator, unitors, and symmetric braiding (with coherence diagrams given in full detail, e.g., pentagon and hexagon, in [1901.10707]).
- The closed structure is realized by the adjunction:
  $$
  \mathrm{DblCat}(A\otimes B, C) \cong \mathrm{DblCat}(A, [B,C])
  $$
  ([1901.10707]). This structure ensures that various double category constructions (e.g., identity, monads, quintets) are strongly monoidal functors.

## 7. Applications and Significance

Symmetric monoidal double categories provide the essential setting for advanced categorical constructions, particularly where "two-dimensional" morphisms interact with symmetric tensor products. Notable areas of application include:
- **Modular Functors and Topological Field Theories:** Modular functors arising from string-net and skein-theoretic constructions are framed as symmetric monoidal double functors out of bordism double categories ([2605.03708]).
- **Functorial Constructions in Higher Category Theory:** The passage from double categories with symmetric monoidal structure to symmetric monoidal bicategories underpins modern approaches to higher categorical algebra and topological quantum field theory ([1910.09240], [1004.0993]).
- **Abstract Categorical Frameworks:** The double categorical perspective unifies and extends classic categorical objects, such as spans, profunctors, operadic bimodules, and Markov processes, supporting a wide variety of algebraic and geometric applications ([2511.14402]).

Symmetric monoidal double categories, equipped with precise lifting properties and internal monoidal structures, thus constitute a central organizing principle in higher-dimensional algebra and its applications.

Source: https://www.emergentmind.com/topics/symmetric-monoidal-double-category