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Stream: theory: philosophy

Topic: adjunctions and dualities

Antonio Lorenzin (May 21 2025 at 11:32):

In the last few years, I've developed a certain idea/opinion/bias that may be worth putting out there and having challenged by the community at large. In my view, a good way to motivate adjunctions is through the concept of duality, which permeates our way of thinking about mathematical objects (from very basic ones, like the relationship between closed and open sets, to higher level abstractions, like scheme theory). 

Let me point out that whether contravariant equivalences are sufficient to encode dualities in general is debatable, as there are counterexamples to such an idea, for example Isbell duality.  So, my claim is that adjunctions are associated with a "relaxed" version of duality (or, perhaps, the right version of it). This leads me to believe that the centrality of adjunctions in mathematics may stem from the fact that they embody a form of duality that "makes more sense": strong enough to be useful and meaningful in many contexts, yet weak enough to be flexible and widely applicable.

In particular, I feel that duality deserves more attention when motivating adjunctions in mathematical discussions, especially since it is more familiar and intuitive for those coming from outside category theory.

David Corfield (May 21 2025 at 11:51):

In line with Urs Schreiber's thinking, e.g., here:

image.png

David Corfield (May 21 2025 at 11:54):

There's an article of mine you might enjoy:

• 2017, Duality as a category-theoretic concept, Studies in History and Philosophy of Modern Physics, Volume 59, August 2017, Pages 55-61, (pdf)

Alex Kreitzberg (May 21 2025 at 14:24):

Lambek had a more specific view, that you could think of Adjunctions as "an amusing analogy" for the unity of opposites. He talked about this in his paper "The influence of Heraclitus on modern mathematics", as well as noting that Lawvere saw an opportunity to make this philosophical idea fully precise.

Lawvere's approach for modeling unity of opposites involves three functors with two adjunctions(a nice stack exchange answer on this: https://math.stackexchange.com/questions/2357569/can-you-explain-lawveres-work-on-hegel-to-someone-who-knows-basic-category-theo)

Lambek was fine with just one adjunction.

I don't know enough philosophy to speak on the relative merits of their approaches.

David Corfield (May 21 2025 at 15:03):

Alex Kreitzberg said:

Lawvere's approach for modeling unity of opposites involves three functors with two adjunctions

The nLab talks about that at [[adjoint modality]].

Of course, an [[adjoint triple]] is a kind of adjunction of adjunctions:

image.png

Alex Kreitzberg (May 21 2025 at 15:31):

I struggle with the "primordial ooze" of category theory. After the tenth "A this is just a that" I stopped trying to explain things in terms of other things and just try to keep track of mental pictures for each widget.

But that feels somehow like the opposite of what you're supposed to do in category theory :laughing:.

Maybe there's an adjunction between understanding just one thing and understanding everything related to one thing. (As I say that as a joke I immediately wonder "is that a, too weak, statement of the yoneda lemma?" This subject never lets me relax!)

John Baez (May 21 2025 at 18:28):

After the tenth "A this is just a that" I stopped trying to explain things in terms of other things and just try to keep track of mental pictures for each widget.

I think it's good to develop a mental image of each widget and use these to see why various "a this is just a that" facts are true. For example I can see why a monoid object in the bicategory of spans of sets is just... a category! At first this seemed like a very 'abstract' and scary fact, which was easy to forget, but now I can just see it.

(I'm using visual metaphors for understanding because I do in fact have a mental image of what's going on here, but verbal, non-visual understanding is also powerful and can be better.)

Kevin Carlson (May 21 2025 at 19:03):

I'm curious whether the mental image is an image of a span, eg of the actual glyphs $O\leftarrow M\to O$ as you'd write them out to check this result, or an image of some particular elements of this span.

John Baez (May 21 2025 at 20:08):

The notation $O \leftarrow M \rightarrow O$ is a nice instance of the microcosm principle, since if these are sets I see each one as a bag of dots, and each dot in $M$ gets mapped to two dots in $O$ thus:

$\bullet \leftarrow \bullet \rightarrow \bullet$

though in my imagery it's important that those arrows point diagonally down.

Matteo Capucci (he/him) (May 27 2025 at 07:35):

David Corfield said:

Alex Kreitzberg said:

Lawvere's approach for modeling unity of opposites involves three functors with two adjunctions

The nLab talks about that at [[adjoint modality]].

Of course, an [[adjoint triple]] is a kind of adjunction of adjunctions:

image.png

is the opposite true? is any adjunction of adjunctions of that form?

David Corfield (May 27 2025 at 09:59):

Yes, in the sense that there's an equivalence:

image.png

Oh, was the question whether there are cases of

$(F \dashv G) \dashv (J \dashv H)$

with $G$ and $J$ distinct?

Matteo Capucci (he/him) (May 30 2025 at 15:01):

exactly!

Morgan Rogers (he/him) (May 30 2025 at 19:06):

I think the 2-cells between adjunctions are natural transformations between the component functors. That would mean that an adjunction between the adjunctions makes the involved functors adjoint also. In other words we would have $(F \dashv J)$, which makes $G \cong J$ by essential uniqueness of adjoints.