Timeline for Universal Objects in Big Categories
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 5, 2011 at 18:59 | vote | accept | QcH | ||
| May 5, 2011 at 18:03 | vote | accept | QcH | ||
| May 5, 2011 at 18:59 | |||||
| May 5, 2011 at 18:03 | comment | added | QcH | Great! This clarifies a lot of things! Thanks a lot. | |
| May 5, 2011 at 18:00 | comment | added | Todd Trimble | @Qph: yes, you're right that such "big" (non-locally-small) categories sometimes arise, but this is not an issue. I think Emil has already answered you, but I'll say it in slightly different words: for any first-order formula $\phi$ with one free variable $x$, $\exists ! x \ \phi(x)$ is shorthand for the sentence $\exists x \ \phi(x) \wedge \forall x, y \ (\phi(x) \wedge \phi(y) \Rightarrow x = y$. | |
| May 5, 2011 at 17:52 | comment | added | QcH | @Pete L. Clark: I was asking a general question related to big categories, and $\mathcal{Sets}$ is just an example demonstrating my question. | |
| May 5, 2011 at 17:39 | comment | added | Pete L. Clark | @Qph: I don't understand your comment: the collection of functions between any two sets is a set. | |
| May 5, 2011 at 17:37 | comment | added | QcH | Thanks a lot for your answer! What happens if $\hom(A,B)$ may be proper classes as well when $A$ and $B$ are arbitrary objects? Then, does $\exists!$ still make sense? This is probably a dumb question, but I don't really know what is or what is not allowed when the domain of quantification is not a set. Thanks!!! | |
| May 5, 2011 at 15:40 | history | edited | Todd Trimble | CC BY-SA 3.0 |
added 67 characters in body
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| May 5, 2011 at 15:31 | history | answered | Todd Trimble | CC BY-SA 3.0 |