Timeline for Examples of non-adjoint equivalences
Current License: CC BY-SA 4.0
22 events
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| Jan 27, 2022 at 8:34 | comment | added | Maxime Ramzi | @MikeShulman (and Tim) : I think the situation Tim is referring to also happens quite often in $(\infty,1)$-categories, where specifying a functor is not as easy as just "objects and morphisms", and so very many equivalences are really only obtained by exhibiting an antecedent - of course, you could say what it does on objects and morphisms, but to specify the full inverse functor (and the isomorphisms) you just invoke the existence of inverses | |
| Jan 26, 2022 at 22:11 | history | edited | Alec Rhea |
edited tags
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| Jan 26, 2022 at 21:50 | history | edited | Paul Taylor |
no need for triangles or lists
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| Jan 26, 2022 at 21:49 | answer | added | Paul Taylor | timeline score: 4 | |
| Jan 26, 2022 at 19:26 | comment | added | Mike Shulman | Yes, canonicality is irrelevant to the original question! But I do think a left adjoint equivalence is automatically also a right adjoint equivalence: just invert and exchange the unit and counit. | |
| Jan 26, 2022 at 18:29 | comment | added | Tim Campion | I think this discussion of what "canonical" means (where I think the main point on which we're talking talking past each other is "what kind of auxiliary data is 'allowed'") is immaterial to the question at hand. Any theorem of the form "$F$ is an equivalence" will produce some unit and counit, "canonical" or no. The question can be interpreted as "what are some examples where the unit and counit handed to us by the theorem are non-adjoint". Right? | |
| Jan 26, 2022 at 18:20 | comment | added | Tim Campion | It also deserves to be said that a left adjoint equivalence will generally not be a right adjoint equivalence. | |
| Jan 26, 2022 at 16:47 | comment | added | Reid Barton | @MikeShulman I misunderstood your intent as "once you fix a construction of (co)limits, the entire package of the equivalence is canonical", but now I see you meant something else. // In any case, examples constructed this way won't be examples of non-adjoint equivalences. | |
| Jan 26, 2022 at 16:00 | comment | added | Mike Shulman | One mathematical difference between constructions involving (co)limits and those like Tim's is that (co)limits can be constructed (albeit in many ways, with no one of them canonical) without the axiom of choice (or even the law of excluded middle). | |
| Jan 26, 2022 at 15:58 | comment | added | Mike Shulman | @ReidBarton Did I say something that sounded like claiming a particular construction of (co)limits was "canonical"? | |
| Jan 26, 2022 at 15:46 | comment | added | Reid Barton | That's true, but I don't see a philosophical justification for treating one construction of (co)limits as "canonical", while not likewise treating one construction of promoting an ess surj+ff functor to an equivalence as "canonical". | |
| Jan 26, 2022 at 15:41 | comment | added | Mike Shulman | @ReidBarton However, once you fix a particular construction of a (co)limit to give the action on objects of the inverse functor, all the rest of the data is determined canonically. | |
| Jan 26, 2022 at 15:40 | comment | added | Mike Shulman | @TimCampion It's true that there are artificial examples like that, but I don't think they arise very often in nature. | |
| Jan 26, 2022 at 15:02 | comment | added | Reid Barton | That's how I interpreted your comment, Tim, but maybe worth adding that even the "inverse" functor is selected "at random", and not just the unit/counit. A more typical example is where the inverse functor is produced by constructing (co)limits. Of course we can give a general construction of (e.g.) a colimit of sets, but there isn't anything particularly canonical about one construction over all the other ones. | |
| Jan 26, 2022 at 13:24 | comment | added | Tim Campion | @MikeShulman The kind of example I have in mind is the inclusion functor from the category of cardinals with all functions between them, to the category of sets with all functions between them. One shows that this functor is fully faithful and essentially surjective. Unwinding the proof, sure one obtains an inverse functor, a unit, and a counit, but all of this data is dependent on a choice of well-ordering of each set. It's canonical with respect to some auxiliary data (isn't everything in math canonical with respect to auxiliary data?), sure, but in an important sense it's not canonical. | |
| Jan 26, 2022 at 9:53 | comment | added | NameNo | Another way to look at the problem: katmat.math.uni-bremen.de/acc/acc.pdf (Abstract and concrete categories) Remark 5.13 pag. 67 says that when a concrete equivalence from A to B exists, a concrete one from B to A might not exist. Btw. I disagree with the statement "it makes little sense to say that they are concretely equivalent" (for a pair of concrete categories). In 1993 I defined and used a (symmetric) concept of concrete equivalence more general than concrete isomorphism (and less general than the one in the book). | |
| Jan 26, 2022 at 8:34 | comment | added | NameNo | As implicit in previous comments: when you see "equivalent categories" as "having isomorphic skeletons", you hit the big problem of AC and "un-naturality". Avoid it with "having isomorphic inflations": Freyd - Scedrof. I would look at the papers that introduced anafunctors; I cannot acces them in this moment but I am quite confident that the examples they give might also answer this question. (PS: category theory is the only place in mathematics where to define "natural transformation" you use a un-natural definition, and it works. Mostly, rare exceptions like this one). | |
| Jan 26, 2022 at 6:48 | comment | added | Alec Rhea | I tend to agree with Mike — the proof that ff+eso iff antiparallel functors and natural isos provides a formula for getting a unit and counit out of a ff+eso functor, and as Mike points out if we have the eso part explicitly given we have an adjoint equivalence determined. | |
| Jan 26, 2022 at 6:27 | comment | added | Mike Shulman | @TimCampion I'm not sure about that. In my experience a lot of equivalences are constructed by giving two functors and two natural isomorphisms. And even when it's phrased as a proof of ff+eso, the proof of eso generally proceeds by constructing for each $y$ a specific $x$ and a specific isomorphism $fx\cong y$, which then canonically determines the structure of a functor on $y\mapsto x$ and a unit and counit for an adjoint equivalence. | |
| Jan 26, 2022 at 3:42 | comment | added | Reid Barton | If I give a non-adjoint equivalence that means that I've given an equivalence $F$ together with a nontrivial natural automorphism of $F$, right? That should suggest that this situation is uncommon, and might suggest where to look for an example... | |
| Jan 26, 2022 at 3:32 | comment | added | Tim Campion | It deserves to be said that lots of examples of equivalences don't really come with a canonical unit / counit at all -- you just select one at random after showing that a functor is full, faithful, and essentially surjective. | |
| Jan 26, 2022 at 2:10 | history | asked | Alec Rhea | CC BY-SA 4.0 |