Timeline for Is there a good general definition of "sheaves with values in a category"?
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| Nov 23 at 23:37 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Apr 6, 2021 at 15:00 | comment | added | Ivan Di Liberti | @SimonHenry's construction is connected to the tensor $\mathcal{A} \boxtimes T$, as discussed in Rem. 5.2.6 of my thesis or Rem 3.16 of "General facts on the Scott Adjunction". | |
| Apr 6, 2021 at 13:02 | comment | added | Zhen Lin | That is certainly an interesting construction! It shows that my list doesn't capture "cohesiveness": by going via the category of points, for example Hausdorff spaces become identified with their underlying set of points, which is definitely not desirable. Hmmm... | |
| Apr 6, 2021 at 12:27 | comment | added | Simon Henry | Just as an interesting example, here is a "definition" that fits all your criteria: As discussed, wanting copower force filtered colimits to exists. Assuming that $\mathcal{A}$ has filtered colimits, then you can define for a topos $T$, $Sh(T,\mathcal{A})$ as the category of functors $Pt(T) \to \mathcal{A}$ that preserve filtered colimits (where $Pt(T)$ is the category of points of $T$, which always have directed colimits). This satisfies all your requirement (with restriction on (4)). Moreover, if finite limits and filtered colimit comute in $\mathcal{A}$, then the $f^*$ are left exact. | |
| Apr 6, 2021 at 10:55 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Apr 6, 2021 at 10:05 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Apr 5, 2021 at 18:28 | history | became hot network question | |||
| Apr 5, 2021 at 17:18 | comment | added | fosco | I have a bad feeling about this question (although I like it). It's like chasing the elephant without a picture of the elephant. You might end up framing a tiger instead. Why isn't 5 replaced, more generally, by "the category of sheaves on a discrete category $X$, with discrete topology, is canonically equivalent to an $X$-fold product of copies of $\cal A$? | |
| Apr 5, 2021 at 15:47 | comment | added | Simon Henry | Ah I see, the definition I've given gives you functoriality in the $f_*$ direction, and you want functoriality in the $f^*$ direction. I don't think that is possible in general. | |
| Apr 5, 2021 at 15:18 | answer | added | Mike Shulman | timeline score: 22 | |
| Apr 5, 2021 at 14:10 | comment | added | Simon Henry | I started writting some details in an answer because it was too long for a comment. | |
| Apr 5, 2021 at 14:10 | answer | added | Simon Henry | timeline score: 9 | |
| Apr 5, 2021 at 14:00 | comment | added | Zhen Lin | @SimonHenry How do you get contravariant functoriality with respect to geometric morphisms? Specifically, where does sheafification come from if you only assume limits exist? And how do you guarantee that sheafification has good properties? | |
| Apr 5, 2021 at 13:57 | comment | added | Simon Henry | I'm not sure I agree with your claim that the "naive" way of defining sheaves in a category does nto give you these properties. As long as you restrict to category with all limits they are essentially all satisfied (excepte maybe 1 that is a very vague). Of course, trying to defines sheaves with value in a category that does not have limits is probably a bad idea as the definition involve plenty of limits. | |
| Apr 5, 2021 at 13:52 | answer | added | Ivan Di Liberti | timeline score: 9 | |
| Apr 5, 2021 at 10:59 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Apr 5, 2021 at 10:25 | history | asked | Zhen Lin | CC BY-SA 4.0 |