Timeline for Origin and context of adjunctions inducing equivalences between full subcategories
Current License: CC BY-SA 4.0
22 events
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| Mar 7, 2021 at 0:23 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 6, 2021 at 15:56 | answer | added | user175321 | timeline score: 2 | |
| Mar 6, 2021 at 12:49 | comment | added | user175321 | Thanks to MO's "Related"-feature I realized that this question has already been answered here: mathoverflow.net/questions/36766/… (2/2) | |
| Mar 6, 2021 at 12:49 | comment | added | user175321 | In a previous version of my post I asked whether the theorem above can be formulated as an equivalence between the category of algebras over the monad induced by the adjunction and the category of coalgebras over the comonad induced by the adjunction (this was motivated by the fact that this holds for adjunctions between posets, in which case the algebras and coalgebras turn out to be the same as the fixed points $P'$ and $Q'$). (1/2) | |
| Mar 6, 2021 at 12:46 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 6, 2021 at 12:36 | comment | added | user175321 | @PaulTaylor I see, thanks. However, given the adjunction between topological spaces and locales and the goal to restrict this adjunction to an equivalence between full subcategories, how do you get the idea to consider sober spaces and spatial locales "by hand", i.e., without knowing the above theorem. Isn't the whole task much easier when one knows that it's useful to consider the objects for which the unit/counit is an isomorphism? | |
| Mar 6, 2021 at 12:04 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 6, 2021 at 11:56 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 6, 2021 at 11:22 | comment | added | user175321 | @MarcOlschok Indeed, thanks! | |
| Mar 6, 2021 at 7:28 | comment | added | David Roberts♦ | Maybe work of Kan? He did discover adjunctions, after all. | |
| Mar 6, 2021 at 0:36 | comment | added | Marc Olschok | The case of posets is in CWM IV-5, Exercise 2. Part of the exercise is the question of a generalization to arbitrary adjunctions. So the author was aware of the theorem but assumed that it is more fun for the reader to work it out independently. | |
| Mar 5, 2021 at 22:58 | comment | added | Paul Taylor | OK, as a useful exercise for a new grad student, run through the adjunctions that you know and characterise their equivalent subcategories. My guess is that those where the two categories are "approximately the same idea", eg topological spaces and locales, will have big equivalent subcategories, but other situations will yield empty ones. | |
| Mar 5, 2021 at 22:42 | comment | added | user175321 | @PaulTaylor Thanks. But I don't agree that just because a statement has an easy or even trivial proof it is a poor theorem. (Your comment somehow suggests this viewpoint.) | |
| Mar 5, 2021 at 22:21 | comment | added | Paul Taylor | The "Theorem" is an obvious corollary of the triangle equations, which are certainly in CWM. A random adjunction will restrict to quite a small, probably empty, equivalence. The situations where there is an interesting one are probably obvious from the outset. So I am not sure that there is any interesting "Theorem" that CWM omitted. Expressing it all in modern abstract language gets the whole question upside down from a historical perspective. | |
| Mar 5, 2021 at 20:31 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 5, 2021 at 19:40 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 5, 2021 at 19:31 | comment | added | user175321 | That's remarkable: I just looked into Mac Lane's CWM for reference of the above theorem but it seems it isn't in there! | |
| Mar 5, 2021 at 19:28 | comment | added | user175321 | Googling "profunctor" I haven't found a text that explains how profunctors induce adjunctions in the same way that relations induce Galois connections. Concerning 3: I haven't computed any examples. My hope is that the answer to this question is well-known among category theorists. Concerning 4: I agree that it is a soft question. But it would be helpful if an expert could just tell me "I used the theorem to prove an equivalence result I wouldn't have discovered without the theorem" or "In my experience nearly all equivalence results arise without the theorem". | |
| Mar 5, 2021 at 18:25 | history | edited | user175321 | CC BY-SA 4.0 |
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| Mar 5, 2021 at 14:25 | comment | added | Kevin Carlson | Not sure about 1. For 2, lookup "profunctor". For 3, have you computed any examples? 4 seems too vague to answer. | |
| Mar 5, 2021 at 12:05 | review | First posts | |||
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| Mar 5, 2021 at 12:00 | history | asked | user175321 | CC BY-SA 4.0 |