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I added some references to adjoint triple for the folklore theorem about fully faithful adjoint triples.
Several lemmas concerning adjoint pairs and adjoint triples are included in
Unfortunately there is a somewhat nonstandard usage of terminology continuous functor (and flatness there includes having right adjoint).
Feel free to add them to the page! (-:
I added to adjoint triple a sketch proof that one of the adjunctions is an idempotent adjunction if and only if the other is, somewhat analogously to the situation with full-faithfulness.
added the beautiful observation by Dan Licata and Mike Shulman, that adjoint triples are equivalently adjunctions of adjunctions.
So is an adjoint quadruple, like we find in cohesion, an adjoint triple in adjunctions? Presumably yes, as an adjoint quadruple is a pair of adjoint triples, so could be arranged as two squares vertically.
But then is it also an adjunction in the 2-category of adjoint triples? (Arranging the squares horizontally.)
Maybe most entertainingly, an adjoint quadruple should be an adjunction of adjunctions of adjunctions. A third order duality.
So a cube.
Thanks! I modified it a bit so that it doesn’t appear to be claiming that we discovered that fact.
This mode/adjoint logic is reminding me of Neel Krishnaswami’s comment.
I wonder if Peirce’s gamma graph constructions have anything to teach us still. Given that Peirce provided you, Mike, with a diagrammatic notation for monoidal fibrations, it’s not impossible. For a small taste, on p. 24 of this we hear about his use of tinctures to depict “As far as is known, a Turk exists who is the husband of two different persons”.
I see I’ve been promoting this for a while:
And once you’ve sorted out Peirce’s beta system, I look forward to a category theoretic rendition of the gamma system.
By the way, the reference [12] has a misspelt ’cirlce’.
Dotting the i’s in the reference section: in [7] it’s officially ’F. W. Lawvere’ I think and for my taste the capital letters of Mac Lane in [10] and Meyer-Vietoris in [4] should be protected as well (ultimately this depends upon the exact titles of the papers cited though).
Slightly more relevant: Are you aware of this arXiv-paper ? It has certain parallels with your approach.
Another thing: what these guys (doi:S0304-3975(98)00359-4) are doing might turn out to be an instance of the adjoint logic with a quintessential localization as mode category or at least this might suggest a direction for possible applications of such a gadget.
You guys need to be explicit that you are giving feedback on my paper with Dan! At first I thought you were talking about the references in the nLab article (which this discussion page is, after all, about). Thanks for the suggestions, I’ll look into them.
On a similar issue of explicitness as #12, was your ’Cute!’ referring to Neel’s comment, Mike?
I guess; that was a long time ago. (-:
Regarding the statement that adjunctions of adjunctions are adjoint triples:
Why the need/desire to define the 2-morphisms in $Adj$ (p. 13) to be “conjugate pairs” of natural transformations?
If we consider instead the 2-category of categories, left adjoint functors, and plain natural transformations between these. Then an adjunction in that 2-category is still/already an adjoint triple.
Those two 2-categories are biequivalent, so up to equivalence it doesn’t matter. And sometimes it’s useful to have both adjoints as given data, e.g. it corresponds more directly to what we see in the type theory.
Added a link to essential geometric morphism.
Thanks for fixing typos.
Just to clarify, since your last edits in the edit history are shown empty (revision 47 and revision 46):
In revision 45 you fixed two typos in the declaration of the composite endomorphisms, in the proof of this Prop.
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