Timeline for Is an objectwise subframe a sub-inf-lattice in a topos?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Oct 20, 2016 at 14:14 | vote | accept | Arrow | ||
| Oct 19, 2016 at 14:46 | history | edited | Arrow | CC BY-SA 3.0 |
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| Oct 19, 2016 at 13:39 | comment | added | მამუკა ჯიბლაძე | @SimonHenry Right, I now realized there indeed are such examples. Probably the simplest one is in $\mathbb Z/2\mathbb Z$-sets. A two element set with a nontrivial involution has underlying set isomorphic to that of $\Omega$ but the latter has a trivial action. Presumably there also exist simple localic examples. | |
| Oct 19, 2016 at 12:43 | comment | added | Simon Henry | Well you can start with a $F$ that is indeed a subobject of $\Omega$ and then mix every $F(X)$ by some chosen automorphism of $\Omega(X)$ for each $X$. You end up with $F(X)$ being identified with a subset of $\Omega(X)$ for all $X$ but not in a way compatible to functoriality.I'm not sure how to construct an example where it is not possible to make the map agree by changing the maps $F(X) \rightarrow \Omega(X)$ but I don't see any reasons for this not being possible | |
| Oct 19, 2016 at 12:28 | comment | added | მამუკა ჯიბლაძე | @SimonHenry What would be a counterexample? That is, a sheaf $F$ with $F(X)\subseteq\operatorname{Sub}(X)$ for all $X$ but induced maps on $F$ not agreeing with those of $\operatorname{Sub}$? And can it happen that $F$ is not even isomorphic to one with agreeing induced maps? | |
| Oct 19, 2016 at 11:30 | answer | added | Simon Henry | timeline score: 3 | |
| Oct 18, 2016 at 23:48 | comment | added | Simon Henry | A first remark: You also need to add this assumption that for every morphism $f :X \rightarrow Y$ the induced morphism $F(Y) \rightarrow F(X)$ corresponds to the pullback when you identify element of $F(X)$ and $F(Y)$ with subobjects of $X$ and $Y$ to deduce that $F$ is a subobject of $\Omega$. For the rest of the question, this is more subtle: roughly being a frame internally imply being a frame on each object, but being a frame internally is considerably stronger (see the famous "An Extension of the Galois Theory of Grothendieck") I'll think more about it tomorrow | |
| Oct 18, 2016 at 23:12 | history | asked | Arrow | CC BY-SA 3.0 |