Timeline for Yoneda on a not so small category
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2013 at 13:13 | vote | accept | wrongfound | ||
| Nov 2, 2013 at 12:31 | comment | added | Paul Taylor | The core of the question was whether you need set theory for the Yoneda lemma. I explained why you don't. Yes, there are lots of powerful corollaries (hence the name "lemma") and re-interpretations in different categorical and set-theoretic formulations, but I think they distract attention from the core point. | |
| Nov 2, 2013 at 11:51 | comment | added | Michal R. Przybylek | This is the weak Yoneda lemma, which is not the same as Yoneda lemma unless you work in a kind of a Grothendieck universe (or a universe that says how internal $e$ interacts with external $\in$). That's what I called "comparing apples with apes". | |
| Nov 2, 2013 at 11:50 | comment | added | Michal R. Przybylek | Paul, I still don't understand you. It seems to me that you are doing exactly the opposite to what you want to achieve. Let me rename the usual set-theoretic membership relation $\in$ to $e$. In your "Yoneda lemma" you may say something like (perhaps, you would prefer to express the below without using meta-set-theoretic notion): $$ \{x \colon x {\;e\;} FA\} \approx \{\theta \colon \textit{$\theta$ is a nat. tr. $h_A \rightarrow F$}\}$$ (cont) | |
| Nov 2, 2013 at 10:20 | history | edited | Paul Taylor | CC BY-SA 3.0 |
made argument more precised and replied to comment.
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| Nov 2, 2013 at 8:42 | comment | added | Michal R. Przybylek | Similarly, Yoneda lemma doesn't work in the context of internal categories, unless your ambient category is locally cartesian closed. | |
| Nov 2, 2013 at 8:42 | comment | added | Michal R. Przybylek | Paul, I don't understand your answer --- it's perhaps too philosophical for me. If people think that they need any kind of Set Theory to speak about Yoneda lemma, then they are wrong. But if you think that you can gain anything in mathematics by waving hands and being imprecise, then you are wrong too. In the context of enriched categories, Yoneda lemma works when you can define an internal object of natural transformations (and it is almost tautological then); otherwise, you have to state the Yoneda lemma on the sections of objects, which is almost never what we actually need. (cont) | |
| Nov 1, 2013 at 20:40 | history | answered | Paul Taylor | CC BY-SA 3.0 |