Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: dinatural transformations

Zanzi (Apr 27 2023 at 21:42):

I'm looking for a setting where I could work with both natural and dinatural transformations. Since dinatural transformations don't compose by themselves, but compose with natural transformations, is there a structure that combines the two in a sensible way?

Emily (Apr 28 2023 at 17:30):

@fosco and I had been working on this around 2020–2021, although we had to stop our work temporarily due to me facing some health issues (these are on-going, although I hope I can get back to being able to work again with Fosco and finish those papers soon).

As it turns out, dinatural transformations are an instance of what we call $W$-weighted natural transformations, which work in the same way as natural transformations $\alpha\colon F\Rightarrow G$ between functors $F,G\colon\mathcal{C}\to\mathcal{D}$, but are weighted with respect to a profunctor $W\colon\mathcal{C}^\mathsf{op}\times\mathcal{C}\to\mathsf{Sets}$. These form a set $\mathrm{Nat}^W(F,G)$, and we have suitable horizontal and vertical composition laws for them, with e.g. vertical composition taking the form of a map

$$\circ\colon\mathrm{Nat}^{W'}(G,H)\times\mathrm{Nat}^{W}(F,G)\to\mathrm{Nat}^{W\times W'}(F,H).$$

This behaves like the multiplication in a graded ring, where ordinary natural transformations are "degree 0" and $W$-weighted ones are "degree 1", and so you can precompose or postcompose weighted natural transformations with ordinary ones, but when you compose two weighted natural transformations (weighted by $W$ and $W'$, say) you get something in "degree 2", a $(W\times W')$-weighted natural transformations.

Now, dinatural transformations turn out to be exactly the $W$-weighted natural transformations with $W=\mathrm{Hom}_\mathcal{C}$, and thus we end up with composition laws for these as well as a special case! In this case, dinatural transformations pre- or postcompose with natural transformations, but you can also compose dinaturals with dinaturals, obtaining a "$(\mathrm{Hom}_{\mathcal{C}}\times\mathrm{Hom}_{\mathcal{C}})$-weighted natural transformation", and so on. You can then put together natural transformations, dinaturals, $(\mathrm{Hom}_{\mathcal{C}}\times\mathrm{Hom}_{\mathcal{C}})$-weighted natural transformations, $(\mathrm{Hom}_{\mathcal{C}}\times\mathrm{Hom}_{\mathcal{C}}\times\mathrm{Hom}_{\mathcal{C}})$-weighted natural transformations, and so on into an $\mathbb{N}$-graded ring :)

Zanzi (Apr 28 2023 at 22:03):

Oh, that's super cool! So you've actually solved the problem of dinatural transformations not composing? That's really interesting!

Zanzi (Apr 28 2023 at 22:03):

So the specific setting that I'm working in is the proarrow equipment of categories, functors, profunctors, and natural transformations. It doesn't quite capture everything that I'm interested in though, since dinatural transformations come up a lot, and I don't know how to work with them in this setting. Would you have some advice for incorporating them here? I think the full generality of weighted natural transformations is a bit too much for me at the moment (though I imagine coming back to it later).

Nathanael Arkor (Apr 28 2023 at 22:10):

What's an example of something you're trying to do using proarrow equipments that involves dinatural transformations?

Mike Shulman (Apr 28 2023 at 22:17):

In particular, are you sure you want fully general dinatural transformations rather than just [[extranatural transformations]]?

Zanzi (Apr 28 2023 at 23:34):

@Nathanael Arkor they come up a lot in functional programming, ie eval : (a → b, a) → b, or fold: (H a → a) → μH → a

Zanzi (Apr 28 2023 at 23:34):

@Mike Shulman hmm, that actually looks spot-on. so if i understand correctly, the right settings for working with extranatural transformations are compact double categories, or compact closed proarrow equipments?

Emily (Apr 29 2023 at 00:09):

Zanzi said:

Oh, that's super cool! So you've actually solved the problem of dinatural transformations not composing? That's really interesting!

I'm really happy you find it exciting! :grinning_face_with_smiling_eyes:

Emily (Apr 29 2023 at 00:09):

Zanzi said:

So the specific setting that I'm working in is the proarrow equipment of categories, functors, profunctors, and natural transformations. It doesn't quite capture everything that I'm interested in though, since dinatural transformations come up a lot, and I don't know how to work with them in this setting. Would you have some advice for incorporating them here? I think the full generality of weighted natural transformations is a bit too much for me at the moment (though I imagine coming back to it later).

hmmm, I haven't thought about how the 'weighted category theory' framework Fosco and I are developing interacts with proarrow equipments. Could you give some more details on what you're working on? Like in the example you gave to @Nathanael Arkor, how do proarrow equipments relate/apply to eval and fold (or what is your general idea in relating these two)?

Mike Shulman (Apr 29 2023 at 04:50):

Zanzi said:

so if i understand correctly, the right settings for working with extranatural transformations are compact double categories, or compact closed proarrow equipments?

Arguably, yes. But there isn't a really satisfactory definition of those things in the literature yet.