Timeline for Universal Objects in Big Categories
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 6, 2011 at 22:02 | history | edited | QcH | CC BY-SA 3.0 |
added 15 characters in body; added 24 characters in body
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| May 5, 2011 at 18:59 | vote | accept | QcH | ||
| May 5, 2011 at 18:04 | comment | added | QcH | I think it's clearer now. Thanks a lot (for being patient with me -- my knowledge about logic is very limited). | |
| May 5, 2011 at 18:03 | vote | accept | QcH | ||
| May 5, 2011 at 18:59 | |||||
| May 5, 2011 at 17:52 | comment | added | Todd Trimble | @Qph: I just saw your edit. Why should anything change? The treatment given in the answer by Emil or by me is the same whether or not $\hom(A, B)$ is small or not. (As mentioned in these answers, a class in ZFC is given by [is informally the extension of] an unary predicate written in ZFC, and thus you can manipulate them as you do predicates. Insofar as they may not be sets, some set-theoretic constructions like power sets may not be applicable to them, but nothing like that applies to your question.) | |
| May 5, 2011 at 17:47 | comment | added | Emil Jeřábek |
This does not make any difference. Judging from your comment below, your worry is about the $\exists!$ quantifier. However, $\exists!x\,A(x)$ is a short hand for $\exists x\,\forall y\,(A(y)\leftrightarrow x=y)$ (or something equivalent), it does not involve the cardinality of any sets (or classes).
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| May 5, 2011 at 17:38 | comment | added | QcH | Thanks a lot! How about when the collection of arrows are also proper classes? (See EDIT 1 above). | |
| May 5, 2011 at 17:33 | history | edited | QcH | CC BY-SA 3.0 |
added 142 characters in body
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| May 5, 2011 at 15:31 | answer | added | Todd Trimble | timeline score: 10 | |
| May 5, 2011 at 15:27 | comment | added | Emil Jeřábek |
That is, $t$ is a terminal object iff it satisfies the formula $(\forall u)(O(u)\to(\exists!f)H(f,u,t))$, where $O(x)$ and $H(f,x,y)$ are formulas defining the classes of objects and morphisms of $\mathcal A$, respectively.
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| May 5, 2011 at 15:14 | comment | added | Emil Jeřábek | In first-order logic, you can quantify over all objects of the given theory (in this case, ZFC). There is no reason whatsoever why quantification should be restricted to a set. That’s a basic principle of first-order logic. | |
| May 5, 2011 at 15:10 | history | asked | QcH | CC BY-SA 3.0 |