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Stream: theory: category theory

Topic: What is the counit of the Isbell duality adjunction?

Emily (Apr 14 2024 at 03:11):

I'm trying to figure out an elementary explicit expression for the counit

$$\epsilon \colon \mathsf{O}\circ\mathsf{Spec}\Rightarrow\mathrm{id}_{\mathsf{CoPSh}(\mathcal{C})}$$

of the Isbell adjunction, which has components of the form

$$\epsilon_F \colon\mathrm{Nat}(\mathrm{Nat}(F,h^{-}),h_{A})\Rightarrow F$$

for $F\in\mathsf{CoPSh}(\mathcal{C})$.

Here $\mathrm{Nat}(\mathrm{Nat}(F,h^{-}),h_{A})$ is the set of natural transformations from

• $\mathrm{Nat}(F,h^{-})$, the presheaf given by $X\mapsto\mathrm{Nat}(F,h^{X})$.

to $h_A$.

Has anyone here worked this out, or know a reference that does so?

Morgan Rogers (he/him) (Apr 14 2024 at 14:06):

What is $A$?

Morgan Rogers (he/him) (Apr 14 2024 at 15:34):

This sounds like fun. If you don't mind, I'll do things with the opposite variance so that the Yoneda embeddings $よ: \mathcal{C} \to \mathrm{PSh}( \mathcal{C})$ and $ヨ : \mathcal{C} \to \mathrm{CoPSh}( \mathcal{C})^{\mathrm{op}}$ (I recently took to using a left-right reflected $よ$ for the latter, but it's not so easy to write here on Zulip, so I used the katakana instead; I don't actually know if that's a sensible thing to do, but at least they're both pronounced "yo"). Now the left adjoint $L: \mathrm{PSh}( \mathcal{C}) \to \mathrm{CoPSh}( \mathcal{C})^{\mathrm{op}}$ sends a representable presheaf $よ(C)$ to $ヨ(C)$ and thus since it must preserve colimits, it must send a generic presheaf $F = \mathrm{colim}_{(x,C) \in \mathrm{Elt}(F)}ヨ(C)$, recalling that the "coYoneda lemma" expresses $F$ as the colimit over its category of elements of the corresponding representables.

To figure out the value of that copresheaf more concretely, we note that a colimit in $\mathrm{CoPSh}( \mathcal{C})^{\mathrm{op}}$ is computed as a limit in $\mathrm{CoPSh}( \mathcal{C})$, but these are computed pointwise! That is, $L(F)(C') = \mathrm{lim}_{(x,C) \in \mathrm{Elt}(F)^{\mathrm{op}}}ヨ'(C)(C')$ (where the prime on the "yo" indicates that we're dealing with covariant representables now), which is $\mathrm{lim}_{(x,C) \in \mathrm{Elt}(F)^{\mathrm{op}}}\mathcal{C}(C,C')$. If you squint at this (or recognize that this expression represents the colimit) you'll see that this corresponds to the set of cocones under the forgetful functor $\mathrm{Elt}(F) \to \mathcal{C}$.

Dually, the right adjoint $R$ sends a copresheaf $G$ to the set of cones over the forgetful functor $\mathrm{Elt}(G) \to \mathcal{C}$. Beware that these are different category of elements constructions! For a presheaf we get a discrete fibration whereas for a copresheaf we get a discrete opfibration.

So what does the unit do? $RL(F)(C')$ is the set of cones with apex $C'$ over the diagram indexed by the collection of all cones under the forgetful functor $U_F:\mathrm{Elt}(F) \to \mathcal{C}$. An element $x \in F(C')$ provides such a cone as follows: or each cocone $\lambda:U \Rightarrow C$, we simply take the component $\lambda_x$; this commutes with all of the required morphisms by definition of morphisms of cocones. Note that this applies even if there were no cocones!

The description of the counit is obtained by dualizing appropriately; the unit may be easier to think about. This gives a more concrete may to understand Isbell self-dual objects (I don't know how much detail is devoted to this in the literature). If $\mathcal{C}$ is a non-trivial category, for the unit at $F$ to be injective there must certainly be at least one cocone under $U_F$, and indeed it is injective iff whenever $x \neq x' \in F(C)$ there is some cocone $\lambda$ with $\lambda_x \neq \lambda_{x'}$. It is surjective if and only if every cone over the diagram of cocones is of the form $(-)_x$ for some $(x,C) \in \mathrm{Elts}(F)$.

Avery and Leinster don't give an explicit description of the reflexive completion (the fixed points of this adjunction) in their paper linked on the nLab page, but I can at least point out that the condition is kind of strange. For instance, if $U_F$ actually has a colimit in $\mathcal{C}$ then the copresheaf $L(F)$ is representable, which means that the double dual also is. Thus such an $F$ is Isbell self-dual if and only if it was already representable!

Morgan Rogers (he/him) (Apr 14 2024 at 16:12):

(Avery and Leinster do mention the Dedekind-MacNeille completion, though, so this is "just" a categorification at that. I was surprised to find that my description also didn't come up in earlier MO discussion )

Morgan Rogers (he/him) (Apr 14 2024 at 16:18):

I guess I should probably tag @Tom Leinster for his paper and @Todd Trimble for his MO answer to see if either of them have seen this 'explicit' description of Isbell duality before.

Peva Blanchard (Apr 14 2024 at 17:13):

I don't know if it helps but there were related discussions on the n-category café:

• Tight spans, Isbell completions, and semi-tropical modules
• Nucleus of a profunctor

Emily (Apr 14 2024 at 17:29):

Morgan Rogers (he/him) said:

What is $A$?

Ahhh sorry! I meant to use $A$ as a dummy variable, with the expression for $\epsilon_F$ reading like $A\mapsto\text{stuff}(A)$. I've fixed it now.

Emily (Apr 14 2024 at 17:46):

Morgan Rogers (he/him) said:

This sounds like fun. If you don't mind, I'll do things with the opposite variance so that the Yoneda embeddings $よ: \mathcal{C} \to \mathrm{PSh}( \mathcal{C})$ and $ヨ : \mathcal{C} \to \mathrm{CoPSh}( \mathcal{C})^{\mathrm{op}}$ (I recently took to using a left-right reflected $よ$ for the latter, but it's not so easy to write here on Zulip, so I used the katakana instead; I don't actually know if that's a sensible thing to do, but at least they're both pronounced "yo").```

Ooh, using ヨ for the contravariant Yoneda embedding is a nice workaround! I use a 180 degrees rotated version, and recently found a hack to use it in MO:

image.png

(Note that here you define the extra-messy expression for \coyo only once, and then you can just use \coyo)

Emily (Apr 14 2024 at 17:51):

It doesn't seem to work on Zulip though =/:

$$\require{HTML} \style{display: inline-block; transform: rotate(180deg)}{よ}$$

fosco (Apr 14 2024 at 20:55):

I like how hiragana and katakana exchange each other under the duality involution $(-)^{コ} : {\bf Jpn} \to {\bf Jpn}$

dusko (Apr 14 2024 at 21:22):

i am not sure whether this is what you are looking for, but the isbell units and counits and the modifications needed to restrict the isbell adjunction to a "dedekind-macneille" completion of categories, are here:
https://arxiv.org/abs/2204.09285

FWIW, let me say that completing ordinary categories (or categories enriched over anything that is not a poset) is a completely different kind of problem from what many of us worked out in the posetal-enriched frameworks, spearheaded by lawvere's generalized metric spaces. on the other hand, the nuclei as spanned by the fixpoints of functors were worked out by lambek in his 1966 Springer Lecture Note #24. this is obviously not complete in any nontrivial sense, and that does not change if the functors are specialized to monads or generalized to profunctors.

Mike Shulman (Apr 14 2024 at 23:34):

I think the katakana ヨ should be avoided. It looks entirely too much like $\exists$.

Morgan Rogers (he/him) (Apr 15 2024 at 05:54):

@Emily the paper that Dusko linked introduces a lot of unconventional notation and is missing some more recent references but it does explicitly contain the presentation of the unit and counit in terms of cones and cocones that I described above, in case you need something more permanent than this Zulip thread :)

Emily (Apr 15 2024 at 20:31):

Morgan Rogers (he/him) said:

Emily the paper that Dusko linked introduces a lot of unconventional notation and is missing some more recent references but it does explicitly contain the presentation of the unit and counit in terms of cones and cocones that I described above, in case you need something more permanent than this Zulip thread :)

Thanks, Morgan!

Emily (Apr 15 2024 at 20:31):

@Morgan Rogers (he/him) Incidentally, since posting this question and reading your reply, I've noticed there's a natural candidate for a simple description of the unit of the Isbell adjunction. Namely:

The unit

$$\eta\colon\mathrm{id}_{\mathsf{PSh}(\mathcal{C})}\Rightarrow[\mathsf{Spec}\circ\mathsf{O}]$$

has components

$$\eta_{\mathcal{F}}\colon\mathcal{F}\Rightarrow\mathrm{Nat}\Big(\mathrm{Nat}\big(\mathcal{F},h_{(-)}\big),h^{(-)}\Big).$$

whose component

$$\eta_{\mathcal{F}|X}\colon\mathcal{F}(X)\to\mathrm{Nat}\Big(\mathrm{Nat}\big(\mathcal{F},h_{(-)}\big),h^{X}\Big)$$

at $X$ takes an element $\phi\in\mathcal{F}(X)$ and sends it to the natural transformation

$$\alpha_{\phi}\colon\mathrm{Nat}\big(\mathcal{F},h_{(-)}\big)\Rightarrow h^{X}$$

having components

$$\alpha_{\phi|A}\colon\mathrm{Nat}\big(\mathcal{F},h_{A}\big)\Rightarrow h^{X}_{A}$$

defined by

$$\alpha_{\phi|A}(\beta) \overset{\scriptstyle\mathrm{def}}{=} \beta_{X}(\phi)$$

for each $\beta\in\mathrm{Nat}(\mathcal{F},h_{A})$ with $X$ component $\beta_{X}\colon\mathcal{F}(X)\to h^{X}_{A}$.

Is this natural transformation the same one as the unit you constructed in your initial reply?

Emily (Apr 15 2024 at 23:31):

Emily said:

I'm trying to figure out an elementary explicit expression for the counit

$$\epsilon \colon \mathsf{O}\circ\mathsf{Spec}\Rightarrow\mathrm{id}_{\mathsf{CoPSh}(\mathcal{C})}$$

of the Isbell adjunction, which has components of the form

$$\epsilon_F \colon[A\mapsto\mathrm{Nat}(\mathrm{Nat}(F,h^{-}),h_{A})]\Rightarrow F$$

for $F\in\mathsf{CoPSh}(\mathcal{C})$.

Here $\mathrm{Nat}(\mathrm{Nat}(F,h^{-}),h_{A})$ is the set of natural transformations from

• $\mathrm{Nat}(F,h^{-})$, the presheaf given by $X\mapsto\mathrm{Nat}(F,h^{X})$.

to $h_A$.

Has anyone here worked this out, or know a reference that does so?

Ah, I just realised I got the direction of the counit wrong here (I wasted so much time because of this...)

$\epsilon$ is a natural transformation between functors landing in $\mathsf{CoPSh}(\mathcal{C})^{\mathsf{op}}$, and not in $\mathsf{CoPSh}(\mathcal{C})$, so its components instead go like

$$\epsilon_F \colon F\Rightarrow[A\mapsto\mathrm{Nat}(\mathrm{Nat}(F,h^{-}),h_{A})].$$

Now I think there's a good chance $\eta$ and $\epsilon$ are both given by evaluation, as I mentioned just above. I'll try to prove this and update this thread in a bit.

Morgan Rogers (he/him) (Apr 16 2024 at 06:05):

That's right, the unit and counit are constructed in essentially the same way.

dusko (Apr 16 2024 at 08:31):

Emily said:

Ah, I just realised I got the direction of the counit wrong here (I wasted so much time because of this...)

$\epsilon$ is a natural transformation between functors landing in $\mathsf{CoPSh}(\mathcal{C})^{\mathsf{op}}$, and not in $\mathsf{CoPSh}(\mathcal{C})$, so its components instead go like

$$\epsilon_F \colon F\Rightarrow[A\mapsto\mathrm{Nat}(\mathrm{Nat}(F,h^{-}),h_{A})].$$

Now I think there's a good chance $\eta$ and $\epsilon$ are both given by evaluation, as I mentioned just above. I'll try to prove this and update this thread in a bit.

maybe you didn't waste time but got some insight into the main question: what do the algebras for the monad induced by the isbell adjunction look like? how do they evaluate the second-order functions? ((they are like the limit operations, say for compact hausdorff spaces))

as for the unit and counit, they are quite simple if you don't use the CWM notation, which is in the meantime more than 50 years old. (if you tried to use van der waerden notation to do algebra simple things would also start looking serious.) e.g., the kan extension units have been used for many years (but less than 50) in continuations and oracles and there they write $\widehat{x}$ for $\eta(x)$. the definition becomes $\widehat{x}(f) = f(x)$.

Emily (Apr 20 2024 at 16:53):

dusko said:

maybe you didn't waste time but got some insight into the main question: what do the algebras for the monad induced by the isbell adjunction look like? how do they evaluate the second-order functions? ((they are like the limit operations, say for compact hausdorff spaces))

I did get the feeling that such a map would feel a bit like a limit somehow from the time I was struggling with trying to come up with one! Now I'm struggling with this question for a couple cases.

One nice result I found is that in the $\mathsf{Vect}_k$-enriched setting, taking $\mathcal{C}$ to be the delooping of $k$ and considering Isbell duality recovers the dual vector space adjunction $(-)^*\dashv(-)^*\colon\mathsf{Vect}_{k}\rightleftarrows\mathsf{Vect}^{\mathsf{op}}_{k}$, and the algebras for this monad are linearly compact topological vector spaces by Theorem 7.8 of Leinster's Codensity and the ultrafilter monad. (PS: I'm sure you're aware of this paper; I'm linking it for the other people who might be following this discussion).

Now I'm trying to figure out a couple other cases, including for $\mathcal{C}$ a poset, $\mathcal{C}$ a one-object category corresponding to a monoid $A$ (which gives an adjunction between left/right $A$-sets) and $\mathcal{C}$ the simplex category (which gives an adjunction between simplicial sets and cosimplicial sets).

(I'm also planning to carefully go through your paper on tight completions in a bit, by the way.)

Do you happen to know any results about what the algebras for Isbell duality might be in these cases, Duško?

dusko (Apr 20 2024 at 20:12):

Emily said:

Do you happen to know any results about what the algebras for Isbell duality might be in these cases, Duško?

well, that is what dominic hughes' and my papers from 2020 are about. i think i left a link to the "tight completions" paper earlier in this thread, and there is also the "nucleus I". both are on arxiv.

we needed to calculate these algebras as the concepts in the latent space of a recommender system (say on a soc network). this is normally done by linear algebra (usually versions of SVD) --- which injects the tacit assumption that the weights are independent, which leads to amplifications (aka the "echo chambers"). but then when you spell out the explicit conditions for the algebras, the problem opens up of effectively recognizing the algebras. ((in recommendations you generate them, but in analysis you really need to recognize them.)) --- so you need to find more effective presentations than the EM-algebras, defined by second-order conditions.

the crucial fact is that the category of algebras for the monad is equivalent with the category of algebras over the category of coalgebras for the comonad. in other words, there is a nuclar adjunction between the category of algebras and the category of coalgebras: right adjoint is monadic whereas the left adjoint is comonadic. that gives you the simple nucleus view of the algebras, as not just effective but also effectively decidable retracts.

that description then displays a symmetry which shows that algebras and coalgebras are much easier to characterize together then separately, and even generate as both limits and colimits, ie tightly. which seems to suggest htat we are retracing the path of statistical inference, just not with numbers but with sets :)

putting together the isbell adjunction with the nuclear adjunction between the induced algebras and coalgebras gives the categorified Latent Semantics. (categorifying is not for the heck of it, but truly necessary to go from dataset to dataset...)

BTW, your approach to the dual pairings through coends is very nice and important. as far as i know, no one worked it out completely. (it was a task, but people got ill and we got stuck before we got to it.)

conceptually, you could think of the dual pairings as a categorical of mackey's dual pairings in TVS... maybe i should continue in your other thread.

dusko (Apr 20 2024 at 20:19):

THANK YOU for asking about this. this is not just a very important application, but also a lot of low hanging fruit... (i am totally NOT giving these speeches about it because "i" was involved with working on this. it is a piece of "music in the air" making each of us smaller than a grain of sand on a beach... or at least me in any case)

Emily (Apr 20 2024 at 22:14):

dusko said:

THANK YOU for asking about this. this is not just a very important application, but also a lot of low hanging fruit... (i am totally NOT giving these speeches about it because "i" was involved with working on this. it is a piece of "music in the air" making each of us smaller than a grain of sand on a beach... or at least me in any case)

Wow, this sounds fascinating! Thank you so much for letting me know about all this!

Emily (Apr 20 2024 at 22:14):

well, that is what dominic hughes' and my papers from 2020 are about. i think i left a link to the "tight completions" paper earlier in this thread, and there is also the "nucleus I". both are on arxiv.

oh, perfect! I'll go carefully go through both then (though I was planning to do so already :)

Emily (Apr 20 2024 at 22:15):

I'll keep working on this thread of dual pairings and see if I can figure out anything interesting; again, thank you so much, Duško =)