Timeline for Do stalks see epimorphism of stacks?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2021 at 21:45 | vote | accept | curious math guy | ||
| Feb 26, 2021 at 17:07 | comment | added | Mike Shulman | I'm not necessarily saying you were wrong to change the question. This version doesn't look to me like the same question you started with, since it says nothing about surjectivity on sections, but I can appreciate that this is probably the question you meant to ask. | |
| Feb 26, 2021 at 17:06 | answer | added | Mike Shulman | timeline score: 6 | |
| Feb 26, 2021 at 16:44 | comment | added | curious math guy | @MikeShulman I honestly just changed the question to make it clearer what I was really asking, it seemed inappropriate to ask a separate question which to my mind was basically the same, but apologies if that was wrong. I will not change this question anymore. | |
| Feb 26, 2021 at 15:09 | comment | added | Mike Shulman | Also, the term "epimorphism" for this notion is arguably inappropriate. In particular, it is not the same as a "monomorphism" in the opposite category. | |
| Feb 26, 2021 at 15:08 | comment | added | Mike Shulman | I'm always wary of posting an answer to a question that keeps changing. Is this your final question? (-: | |
| Feb 26, 2021 at 10:55 | comment | added | curious math guy | @MikeShulman Would you happen to have a reference about this? Also, if you just want to write it quickly into an answer so that this doesn't linger in the open questions part? | |
| Feb 26, 2021 at 2:25 | comment | added | Mike Shulman | Oh, I missed that your previous comment was about the image sheaf surjecting section-wise, unlike the original question. That's true for 1-sheaves and also higher sheaves. In fact, since the image is a subsheaf, it surjects on stalks iff it surjects sectionwise iff it is an isomorphism. | |
| Feb 26, 2021 at 1:47 | comment | added | curious math guy | @MikeShulman, but I'm not claiming anything about section-wise surjections, aren't I? For $1$-sheaves, if the morphism is surjective on stalks, then the morphism of sheaves is an epimorphism, which isn't beeing section-wise surjective, I'm aware of that | |
| Feb 26, 2021 at 1:40 | comment | added | Mike Shulman | No, it doesn't imply that even for 1-sheaves. Consider a topological space $X$ that is the union of two opens $U\cup V$. Then if $y$ denotes the Yoneda embedding, the map of sheaves $yU + yV \to yX = 1$ is surjective stalkwise, but not on sections: $yX$ has a section over the whole space, but $yU + yV$ doesn't. | |
| Feb 26, 2021 at 1:29 | history | edited | LSpice | CC BY-SA 4.0 |
\emph{} -> **
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| Feb 25, 2021 at 23:35 | history | edited | curious math guy | CC BY-SA 4.0 |
Improved the question to make more sense
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| Feb 22, 2021 at 9:57 | comment | added | curious math guy | I'm sorry for my confusion, but for 1-sheaves, surjections on the stalks implies that the image sheaf surjects section-wise. So why does this make it clear that this fails for "higher" sheaves? | |
| Feb 21, 2021 at 15:24 | comment | added | Mike Shulman | To put that differently, if each $\phi_i(f)$ is essentially surjective, then $f$ is internally essentially surjective, i.e. effective-epimorphic. But that doesn't imply that it is sectionwise essentially surjective. That fails already for 1-sheaves: a map of sheaves of sets can be epimorphic without being sectionwise surjective. | |
| Feb 21, 2021 at 12:30 | comment | added | Zhen Lin | A conservative functor reflects anything (not really – but I don't want to be precise here) it preserves. The inverse image functor of a geometric morphism preserves finite limits and (possibly infinite) colimits, so in particular it preserves images, and hence reflects the property of being essentially surjective. But the functor $F \mapsto F (U)$ is not usually the inverse image functor of a geometric morphism. You seem to be confusing "sections" vs "stalks". | |
| Feb 21, 2021 at 9:57 | history | edited | curious math guy | CC BY-SA 4.0 |
deleted 447 characters in body
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| Feb 21, 2021 at 9:55 | comment | added | curious math guy | Oh, right, how embarassing! Apologies, I'll change the question accordingly, since I'm still curious about the essential surjectivity. | |
| Feb 21, 2021 at 2:21 | comment | added | Mike Shulman | What exactly do you mean by an "$n$-sheaf"? In the most common higher-categorical formulation, an $n$-sheaf lives in a weak $(n+1)$-category, where it doesn't even make sense to ask about whether something is an isomorphism (versus an equivalence). | |
| Feb 20, 2021 at 20:19 | history | edited | curious math guy | CC BY-SA 4.0 |
edited title
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| Feb 20, 2021 at 18:38 | history | asked | curious math guy | CC BY-SA 4.0 |