Timeline for is the presheaf category of a locally small category locally small?
Current License: CC BY-SA 2.5
8 events
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| May 14, 2010 at 18:07 | comment | added | user2734 | Thanks for the pointers! The link doesn't seem to work, but I will look at nLab now that I know what to look for. I suppose that at the moment (before finished reading Mac Lane) I'll stick to the foundations described in Section 1.6 of Mac Lane. Although limited (as I now see..), these foundations seem to be sufficient in Mac Lane (as Mike Shulman told me in a comment...), and I should probably not "dive" into something else right now. Thanks again for your help. | |
| May 14, 2010 at 13:28 | comment | added | Todd Trimble | Obviously there's a lot to add, but you're right: these little comment boxes are not the best place. There are various possibilities for "alternative foundations" which avoids these strange glitches. One is ETCS (due to Lawvere), which you can begin reading about at the nLab or from <a href="tac.mta.ca/tac/reprints/articles/11/… himself</a>. (But for some people, ETCS is hard and technical, and not as intuitive as ZFC.) Another very intriguing possibility is Mike Shulman's SEAR (exclusively at the nLab, AFAIK). | |
| May 14, 2010 at 13:09 | comment | added | user2734 | Thank you very much! Having such tips from an expert is extremely helpful. I have a million more question to ask, but I guess these comments aren't the right place for such a mini course... | |
| May 14, 2010 at 12:54 | comment | added | Todd Trimble | Yes, under the sorts of set-theoretic encodings you appear to have in mind, the argument looks to me impeccable but still "morally wrong"! (It would mean $\hom(F, G)$ is not small even if $G$ is terminal.) It's an excellent illustration of the kind of sophistry that's possible by encoding everything as sets and sets as membership trees. More satisfactory would be a foundations where this type of argument cannot even be formulated (cf. discussion of structuralist vs. materialist forms of set theory in the nLab). Suffice it to say that the problem is not insurmountable. :-) | |
| May 14, 2010 at 12:49 | comment | added | user2734 | Thank you very much for your answer! I hope it is OK that I ask another silly question: Is my comment above correct, but just useless, because (for some reason that I still don't understand) "locally small" can refer to non-small hom-sets that are in bijection with small sets? [I'm assuming a single universe, as in Mac Lane.] | |
| May 13, 2010 at 23:39 | comment | added | Todd Trimble | Even if $C$ is large, the collection of transformations $F \to G$ between presheaves can be essentially small (in definable isomorphism with a set). Consider for example $F$ representable and apply Yoneda. | |
| May 13, 2010 at 22:52 | comment | added | user2734 | @Todd Trimble: Is this simple argument wrong? Consider a non-empty hom-set $\widehat{C}(F,G)$, and let $\tau$ be in this hom-set. If the hom-set is small, then by transitivity of the universe, $\tau$ is small too. But $\tau$ is just a function with domain $\operatorname{obj}(C)$ (so $\tau$ is a triple with $\operatorname{obj}(C)$ as its first component). But then from transitivity again we get $\operatorname{obj}(C)\in U$, a contradiction. (So, in general,the answer to the original question is ``no'') | |
| May 13, 2010 at 22:37 | history | answered | Todd Trimble | CC BY-SA 2.5 |