Timeline for Cosheafification
Current License: CC BY-SA 3.0
7 events
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| Mar 20, 2013 at 22:35 | comment | added | arsmath | It's way up there in the hierarchy. Vopenka himself thought it was false -- that's why he proposed it as an argument against the largest of large cardinal axioms -- but I think most set theorists think it's independent. Even if it's fine, it means that you're saying you have no control over the size of cosheafification: it could turn a countable set into a set of cardinality so large that it dwarfs all sets that appear in day-to-day mathematics. | |
| Mar 20, 2013 at 21:13 | comment | added | Justin Curry | @arsmath: Yes, you are right. It has a pretty strong consistency requirement. If I understand correctly, the strength of Vopenka's principle lies between Reinhardt cardinals and unmeasurable cardinals, but I don't have any committed opinions as to how controversial this is. I tend not to worry too much about these things, but perhaps I should. | |
| Mar 20, 2013 at 20:48 | comment | added | Justin Curry | @Ryan: No, a functor $F:C\to D$ always can be used formally to define a functor $F^{op}:C^{op}\to D^{op}$. The assignment of objects remains the same, so $F^{op}(x)=F(x)$, but now a morphism $f:x\to y$ in $C$ becomes a morphism $f^{op}:y\to x$. The functor $F^{op}$ sends $f^{op}$ to $F(f)^{op}$. $F(f):F(x)\to F(y)$ defines a morphism $F(f)^{op}:F(y)\to F(x)$ in $D^{op}$, which is equal to $F^{op}(f^{op}):F^{op}(y)\to F^{op}(x)$. I don't know what you mean by "destroys covers." | |
| Mar 19, 2013 at 14:21 | comment | added | Ryan Reich | You mean $F \colon \mathrm{Open}(X) \to \mathrm{Vect}^\mathrm{op}$, I think. You could alternatively reverse the category $\mathrm{Open}(X)$, but that destroys the open covers and presumably creates more problems. | |
| Mar 19, 2013 at 7:00 | comment | added | arsmath | Your easier-to-check version of the adjoint functor theorem relies on a large cardinal axiom, which is therefore not provable in ZFC unless ZFC is inconsistent. | |
| Mar 19, 2013 at 5:25 | history | edited | Justin Curry | CC BY-SA 3.0 |
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| Mar 19, 2013 at 5:07 | history | answered | Justin Curry | CC BY-SA 3.0 |