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I gave “adjoint cylinder” its own entry.
This is the term that Lawvere in Cohesive Toposes and Cantor’s “lauter Einsen” (see the entry for links) proposes for adjoint triples that induce idempotet (co-)monads, and which he proposes to be a formalization of Hegel’s “unity of opposites”.
In the entry I expand slightly on this. I hope the terminology does not come across as overblown. If it does, please give it a thought. I believe it is fun to see how this indeed formalizes quite well several of the examples from the informal literature.
I am not sure if “adjoint cylinder” is such a great term. I like “adjoint modality” better. Made that a redirect.
Interesting. I think I understand this, at least!
If it says that
$U \;\colon\; \mathbf{L} \dashv \mathbf{R} \,,$generates
$U X \;\colon\; \array{ \mathbf{L} X &\longrightarrow& X &\longrightarrow& \mathbf{R}X \\ opposite\;1 && unity && opposite\;2 }$won’t it be confusing to find
$\int \dashv \flat \,.$generates in the opposite order
$\array{ \flat X \longrightarrow X \longrightarrow \int X ? }$That’s why I added in parenthesis “or the other way around”.
There are two possible kinds of adjoint modalities, namely
$monad \dashv comonad$and
$comonad \dashv monad$Notice that this two-ness is what re-appears in the fact that there are also two choice for adjoint triple modalities, namely the “Yin-triple”
$monad \dashv comonad \dashv monad$and the “Yang-triple”
$comonad \dashv monad \dashv comonad$Sure. But I think it’s better to emphasise the monad/comonad aspect rather than the right/left (which it looks like you’ve gone and done).
Okay, right. I have edited the entry now accordingly.
Also, I edited the piece about the $\flat \dashv \sharp$-adjunction. I see now that this matches well Hegel’s “quantity”
Hegel says in §398
Quantity is the unity of these moments of continuity and discreteness,
Now his “moment of discreteness” is clearly the flat modality. Hence the “moment of continuity” must be the sharp modality, which makes good sense (a sharp modal type is maximally continuous, as every map to it is continuous, by its very definition).
So by Lawvere’s adjoint-cylinder-formalization of unity of opposites, it follows that
$quantity X \;\colon\; \flat X \longrightarrow X \longrightarrow \sharp X$Notice that this neatly matches Lawvere’s interpretation, whose very point in “Cantor’s lauter Einsen” is that this unity of opposite captures Cantor’s Kardinalen.
So we get that Cantor’s Kardinalen corresponds with Hegel’s Quantität, which seems quite right.
What about totally distributive categories?
Thanks for mentioning this! I have added quick cross-pointers to adjoint cylinder/totally distributive category.
The total distributivity of a small category is related to function algebras on infinity-stacks, as it involves the $(\mathcal{O} \dashv Spec)$-adjunction.
BTW, the entry totally distributive category should say why the “waves” there are called “waves”. (Why?)
So $Set$ is unique for having two further left adjoints to make five. Are there any categories with just four?
Is there an $(\infty, 1)$-version of totally distributive?
I should have known/thought of this before. The $\infty$-version of the $Y X$ modality (in the notation at totally distributive category) is what Toën called “affinization” of $\infty$-stacks to affine $\infty$-stacks. But as he showed, what it really is is $\mathbb{A}^1$-cohomology localization.
For the smooth case that’s the content of the entry function algebras on infinity-stacks.
I should say that I don’t know in which cases $affinization = Spec \mathcal{O}$ is the right piece in an adjoint modality. Probably that’s rare. But its a good example of a higher modality, for sure.
They’re called ‘waves’ because of the notation – Johnstone used wavy arrows for them in [Continuous categories and exponentiable toposes].
Is it too late to counteract this bad choice of terminology?
Perhaps. There was a talk at CT2013 all about wavy arrows by Richard Wood, and people seemed completely at home with the word.
I have added a parenthetical remark to the entry explaining the terminology.
I don’t find the terminology “waves” so bad. At least it doesn’t give the wrong intuition. (-:
Are there any categories with just four?
The paper which characterizes Set as having 5 adjoints also proves, I believe, that the categories with 4 are the categories of presheaves on complete lattices.
Does $\infty Gpd$ have 5 $\infty$-adjoints? Or even 4?
I have added in the Idea-section and in the References-section pointers to references on modal logic that consider adjoint modalities.
Also, I renamed the entry from adjoint cylinder to adjoint modality. (Let me know if anyone is opposed to this change.)
I have briefly mentioned at adjoint modality the further example
and added corresponding pointers to formal completion and to torsion approximation. These all just point to the details given at arithmetic fracturing of chain complexes.
It seems very likely that this lifts from chain complexes to spectra, but I don’t see all the details yet. Have asked this on MO, but didn’t ask the question well.
The good thing is that both in chain complexes and in spectra the p-completion is given by derived hom out of something like $\mathbb{Z}/p^\infty$, and the $p$-torsion approximation of course by tensoring with that, but the problem is that the adjunction that this would imply has the wrong variance. Unless I am really mixed up.
added pointer to Lambek’s “The Influence of Heraclitus on Modern Mathematics” which recalls discussion with Lawvere in 1965 about adjunction formalizing dialectics.
(Though unfortunately I have only seen the first two pages of that article by Lambek. If anyone has more pages, I’d be interested.)
added a subsection simple illustrative examples, stated the one about inclusion of even and odd integers that Lawvere suggested, an example of an adjunction of the form $\Box \dashv \bigcirc$; and then added below that the observation that in the same vein there is a simple example of an adjunction of the form $\bigcirc \dashv \Box$, namely floor and ceiling function adjoint to the inclusion of the integers into the reals.
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