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Stream: learning: questions

Topic: Natural transformation of double functors

Peva Blanchard (Nov 17 2025 at 18:20):

Given two double functors $F,G : \mathbb{C} \to \mathbb{D}$ between double categories, how do we define a "natural transformation" from $F$ to $G$?

Maybe I didn't look at the right place, but I couldn't find a definition on the nlab.

Bruno Gavranović (Nov 17 2025 at 18:30):

Definition 12.3.19 in 2-Dimensional categories has it spelled out in great detail.

Peva Blanchard (Nov 17 2025 at 18:46):

Thank you

Kevin Carlson (Nov 18 2025 at 07:35):

This is the most common kind of natural transformation you want to use, but I'd mention just in case that there are at least three very important kinds of morphism between lax double functor, plus more less common ones. The natural transformations as in Johnson-Yau are strictly natural on the tight category, but you can also define a notion that's lax, colax, or pseudo on the tight 2-categories instead, as Definition 7.1 of Cartesian double theories; you can also have pseudo, lax, or colax natural transformations whose components are loose maps (Transposing cartesian structure, Definition 3.1), recovering the more usual kinds of transformations between lax functors of bicategories (but, infamously, not forming a 2-category!), and finally, there are modules between lax double functors (Composition of modules for lax double functors, definition 1.1.6), which generalize profunctors.

Kevin Carlson (Nov 18 2025 at 07:35):

The cited definitions tend to be rather long. It's necessary to understand the unpacked versions for some purposes, but one can also give more memorable short definitions of at least the three most important kinds of morphisms between lax functors. Given $F$ and $G$ as in your post, let $\mathbb{T}=0\to 1$ be the walking tight arrow and $\mathbb{L}=0\nrightarrow 1$ the walking loose arrow. Then a natural transformation (i.e. strict tight transformation) as defined at Bruno's reference is a lax double functor $\mathbb{C}\times \mathbb{T}\to \mathbb{D}$ restricting to $F,G$ on the endpoints. A module between $F$ and $G$ is such a lax double functor $\mathbb{C} \times \mathbb{L}\to \mathbb{D}.$ And if $F,G,$ are themselves pseudo, then a (pseudo) loose transformation is a pseudo functor $\mathbb{C}\times \mathbb{L}\to \mathbb{D},$ appropriately restricting to $F,G$ (if $F$ and $G$ are lax, you can ask for just particular laxators of this functor to be invertible and recover the general definition.)

John Baez (Nov 18 2025 at 12:53):

For many purposes the more you can work with concise definitions like Kevin gave, the better off you'll be!

Peva Blanchard (Nov 18 2025 at 17:35):

Thank you all for the inputs. Indeed, I already grinded my teeth a few times on trying to check the unpacked versions of double categorical definitions. It's good to know compact definitions exist, although I'll need some time to let these sink in.