Timeline for Yoneda lemma for monoidal categories
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 27, 2020 at 2:47 | vote | accept | Jake Wetlock | ||
| Nov 27, 2020 at 2:36 | vote | accept | Jake Wetlock | ||
| Nov 27, 2020 at 2:47 | |||||
| Nov 27, 2020 at 0:13 | answer | added | Mike Shulman | timeline score: 11 | |
| Nov 26, 2020 at 22:23 | history | edited | Jake Wetlock | CC BY-SA 4.0 |
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| Nov 26, 2020 at 19:01 | comment | added | Jake Wetlock | @Mike: Yes, this is precisely the question. I want to know if the assumption is necessary or not. If it is not, I would be much happier :) | |
| Nov 26, 2020 at 17:26 | comment | added | Mike Shulman | I don't understand the question. Is it "why does the nLab page on the enriched Yoneda lemma assume that the monoidal category is locally small?" If so then I think Simon is right that it's not necessary; probably whoever wrote the page was just throwing in a sufficient set of assumptions rather than thinking about which were actually necessary. | |
| Nov 26, 2020 at 16:33 | history | edited | Jake Wetlock | CC BY-SA 4.0 |
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| Nov 26, 2020 at 16:11 | comment | added | Simon Henry | And this is only if one wants $V^C$ to be a $V$-enriched category it self. If one doesn't care that $V^C$ is not always defined, I don't think any size assumption is needed. | |
| Nov 26, 2020 at 16:08 | comment | added | Simon Henry | I would need to check more carefully, but I think you are right: I think the only size assumption are in order for the enriched ends defining $V^C(F,G)$ to exists. But assuming that C has a set of objects, or rather that $V$ has limits of size the set of objects (of isomorphism class actually) of C seem to be enough for this. It does not seems usefull to assume that $C$ or $V$ are locally small. | |
| Nov 26, 2020 at 15:43 | history | edited | Jake Wetlock |
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| Nov 26, 2020 at 15:24 | comment | added | Jake Wetlock | Thanks for the link! I guess what I don't see, or what I am asking, is where in the statement of Yoneda $Hom(Hom(A,-),F \simeq F(A)$ matters of size arise, and why in particular the assumption that $Hom$-sets are small helps in any way. | |
| Nov 26, 2020 at 15:16 | comment | added | varkor | When you take the monoidal category to be Set (which is locally small) with the cartesian product, you recover the usual Yoneda lemma. If you're really interested in the size condition, perhaps the right place to look is into Yoneda structures, which attempt to axiomatise this structure and have to pay close attention to size. | |
| Nov 26, 2020 at 15:14 | comment | added | Jake Wetlock | But in the nLab version they require that the monoidal category is locally small. This seems like a more special version rather than a generalisation. | |
| Nov 26, 2020 at 15:01 | comment | added | varkor | Yes, there is an enriched Yoneda lemma: see this question, or this nLab page. | |
| Nov 26, 2020 at 14:50 | history | asked | Jake Wetlock | CC BY-SA 4.0 |