# nLab monoidal double category

Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

A monoidal double category is a double category equipped with a tensor product (cf. monoidal 2-category), compatible with both tight and loose morphisms.

Formally, a monoidal double category is a pseudomonoid in the (cartesian monoidal) 2-category $\mathbf{DblCat}$ of double categories, (lax) double functors and loose natural transformations?.

## Definition

Recall a double category $\mathbb{D}$ is an internal (pseudo)category in Cat, hence is given by two categories $\mathbb{D}_1$ (category of loose arrows) and $\mathbb{D}_0$ (category of objects), plus source and target functors $s,t:\mathbb{D}_1 \to \mathbb{D}_0$, a loose identity functor $U:\mathbb{D}_0 \to \mathbb{D}_1$, and a loose composition functor $\odot : \mathbb{D}_1 {}_s\times_t \mathbb{D}_1 \to \mathbb{D}_0$. Clearly this data has to satisfy the properties of the composition of a category, thus $U$ provides identities for the $\odot$, which in turn is associative. In Shulman 2010 moreover, these properties are only satisfied up to coherent isomorphism, making loose arrows form a bicategory.

###### Definition

A monoidal double category is a double category $\mathbb{D}$ equipped with double functors

$I: \mathbb{1} \to \mathbb{D}\quad \text{and} \quad \otimes : \mathbb{D} \times \mathbb{D} \to \mathbb{D}$

and with double natural transformations? $\alpha : (\otimes\otimes)\otimes \cong \otimes(\otimes\otimes)$, $\lambda : (I \times \mathbb{D})\otimes \cong \otimes$, $\rho : (\mathbb{D} \times I)\otimes \cong \otimes$ satisfying the usual coherence laws of a pseudomonoid (pentagon and triangle, see monoidal category).

###### Remark

As often happens for structure on internal categories, these arise as structures on the categories of loose arrows $\mathbb{D}_1$ and that of objects $\mathbb{D}_0$ when the source, identity and target maps respect it.

Specifically, a monoidal double category arises when both $\mathbb{D}_1$ and $\mathbb{D}_0$ are monoidal categories and

1. $s,t:\mathbb{D}_1\to\mathbb{D}_0$ are strict monoidal, meaning if $p:A \nrightarrow B$ and $q:A' \nrightarrow B'$ are loose arrows, $p \otimes q$ is a loose arrow $A \otimes A' \nrightarrow B \otimes B'$,
2. $U:\mathbb{D}_0 \to \mathbb{D}_1$ is strong monoidal, meaning $U_1 : I \nrightarrow I$ is the monoidal unit for $\mathbb{D}_1$ and there is an iso $U_{A \otimes B} \cong U_A \otimes U_B$ for all $A,B:\mathbb{D}_0$.
3. $\odot : \mathbb{D}_1 {}_s\times_t \mathbb{D}_1 \to \mathbb{D}_0$ is monoidal, meaning there is an iso $(p \otimes q) \odot (p' \otimes q') \cong (p \odot p') \otimes (q \odot q')$ for any suitably composable loose arrows $p,p',q',q' : \mathbb{D}_1$.

and of course these isomorphisms satisfy coherence axioms one can find in Shulmanβ10.

###### Definition

A braiding on a monoidal double category is a natural transformation $\tau : \otimes \cong \mathrm{swap}\otimes$ which satisfies the coherence laws of the braiding of a pseudomonoid (see braided monoidal category)

###### Definition

A symmetric monoidal double category is a braided monoidal double category whose braiding $\tau$ satisfies $\tau\tau = 1$.

## References

Last revised on September 3, 2023 at 17:08:52. See the history of this page for a list of all contributions to it.