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Stream: theory: category theory

Topic: Dinatural or extranatural transformation, double categories

Patrick Nicodemus (Jul 30 2023 at 00:50):

I have two double categories $\mathbb{A},\mathbb{C}$, a functor $F : \mathbb{A}\to \mathbb{C}$ and a double profunctor $P : \mathbb{A}^{\rm co}\to \mathbb{C}$. I want to define a natural transformation $P\Rightarrow \mathbb{C}(1,F)$ but this doesn't really make sense because of the "co" in $P$. I am wondering if there is a known notion of dinatural transformation or extranatural transformation between functors or profunctors between double categories so I don't have to reinvent the wheel here on figuring out what the reasonable coherence conditions are.

Patrick Nicodemus (Jul 30 2023 at 00:52):

$P$ and $\mathbb{C}(1,F)$ already agree on objects and horizontal arrows, and the natural transformation will be the identity when restricted to these components. The nontrivial part is figuring out the coherence condition on the spans $P(v,w)$ and $\mathbb{C}(v,F(w)$ when $v,w$ are vertical arrows.

Patrick Nicodemus (Jul 30 2023 at 00:54):

Since double categories are just internal categories in Cat, maybe a general concept of a dinatural transformation between functors internal to a category would suffice.

Matteo Capucci (he/him) (Jul 31 2023 at 16:14):

I just happened to stumble upon this nlab page https://ncatlab.org/nlab/show/compact+double+category

Matteo Capucci (he/him) (Jul 31 2023 at 16:14):

See section 4

Patrick Nicodemus (Aug 02 2023 at 17:44):

Oh, cool. Thank you!