nLab monoidal double category



2-Category theory

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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 DblCat\mathbf{DblCat} of double categories, (lax) double functors and loose natural transformations?.


Recall a double category 𝔻\mathbb{D} is an internal (pseudo)category in Cat, hence is given by two categories 𝔻 1\mathbb{D}_1 (category of loose arrows) and 𝔻 0\mathbb{D}_0 (category of objects), plus source and target functors s,t:𝔻 1→𝔻 0s,t:\mathbb{D}_1 \to \mathbb{D}_0, a loose identity functor U:𝔻 0→𝔻 1U:\mathbb{D}_0 \to \mathbb{D}_1, and a loose composition functor βŠ™:𝔻 1 sΓ— t𝔻 1→𝔻 0\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 UU provides identities for the βŠ™\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.


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

I:πŸ™β†’π”»andβŠ—:𝔻×𝔻→𝔻I: \mathbb{1} \to \mathbb{D}\quad \text{and} \quad \otimes : \mathbb{D} \times \mathbb{D} \to \mathbb{D}

and with double natural transformations? Ξ±:(βŠ—βŠ—)βŠ—β‰…βŠ—(βŠ—βŠ—)\alpha : (\otimes\otimes)\otimes \cong \otimes(\otimes\otimes), Ξ»:(I×𝔻)βŠ—β‰…βŠ—\lambda : (I \times \mathbb{D})\otimes \cong \otimes, ρ:(𝔻×I)βŠ—β‰…βŠ—\rho : (\mathbb{D} \times I)\otimes \cong \otimes satisfying the usual coherence laws of a pseudomonoid (pentagon and triangle, see monoidal category).


As often happens for structure on internal categories, these arise as structures on the categories of loose arrows 𝔻 1\mathbb{D}_1 and that of objects 𝔻 0\mathbb{D}_0 when the source, identity and target maps respect it.

Specifically, a monoidal double category arises when both 𝔻 1\mathbb{D}_1 and 𝔻 0\mathbb{D}_0 are monoidal categories and

  1. s,t:𝔻 1→𝔻 0s,t:\mathbb{D}_1\to\mathbb{D}_0 are strict monoidal, meaning if p:A↛Bp:A \nrightarrow B and q:A′↛Bβ€²q:A' \nrightarrow B' are loose arrows, pβŠ—qp \otimes q is a loose arrow AβŠ—A′↛BβŠ—Bβ€²A \otimes A' \nrightarrow B \otimes B',
  2. U:𝔻 0→𝔻 1U:\mathbb{D}_0 \to \mathbb{D}_1 is strong monoidal, meaning U 1:I↛IU_1 : I \nrightarrow I is the monoidal unit for 𝔻 1\mathbb{D}_1 and there is an iso U AβŠ—Bβ‰…U AβŠ—U BU_{A \otimes B} \cong U_A \otimes U_B for all A,B:𝔻 0A,B:\mathbb{D}_0.
  3. βŠ™:𝔻 1 sΓ— t𝔻 1→𝔻 0\odot : \mathbb{D}_1 {}_s\times_t \mathbb{D}_1 \to \mathbb{D}_0 is monoidal, meaning there is an iso (pβŠ—q)βŠ™(pβ€²βŠ—qβ€²)β‰…(pβŠ™pβ€²)βŠ—(qβŠ™qβ€²)(p \otimes q) \odot (p' \otimes q') \cong (p \odot p') \otimes (q \odot q') for any suitably composable loose arrows p,pβ€²,qβ€²,qβ€²:𝔻 1p,p',q',q' : \mathbb{D}_1.

and of course these isomorphisms satisfy coherence axioms one can find in Shulman’10.


A braiding on a monoidal double category is a natural transformation Ο„:βŠ—β‰…swapβŠ—\tau : \otimes \cong \mathrm{swap}\otimes which satisfies the coherence laws of the braiding of a pseudomonoid (see braided monoidal category)


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


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