Timeline for Epimorphisms and 2-isomorphic maps to an algebraic stack
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 18, 2016 at 16:17 | comment | added | stupid_question_bot | Thanks for your comment. I've asked an updated question here: mathoverflow.net/questions/250182/… | |
| Sep 18, 2016 at 7:27 | comment | added | HeinrichD | I wouldn't say that "epimorphism" (either in $1$- or $2$-categorical setting) explains what is going on here. In fact, there is no property at all on $f:T \to S$ which guarantees $$f^* X \cong f^* X' \text{ in } \mathcal{M}(T) \Longrightarrow X \cong X' \text{ in } \mathcal{M}(S),$$ expect for something trivial such as "$f$ is a split epimorphism". This is because $\mathcal{M}(T)$ is the "wrong" category. | |
| Sep 17, 2016 at 22:47 | comment | added | stupid_question_bot | I mean I agree with everything you said in your answer, but can you explain why we "have to work in the category of descent data?" For example, by Yoneda, there's a natural equivalence of categories between $Hom(S,\mathcal{M})$ and $\mathcal{M}(S)$. What you wrote is that $\mathcal{M}(S)\cong\mathcal{M}_{descent}(T\rightarrow S)$, which I understand. So my question effectively becomes: What exactly about my argument involving epimorphisms and universal properties breaks down when we translate everything into (e.g.) descent data? | |
| Sep 17, 2016 at 22:23 | comment | added | HeinrichD | You ask "what's the problem", the answer is "we have to work in the category of descent data, not just in $\mathcal{M}(L)$". Also, I've replaced by $\mathrm{Spec}(L) \to \mathrm{Spec}(K)$ by $f : T \to S$ since fields don't play any essential role here. | |
| Sep 17, 2016 at 22:16 | comment | added | stupid_question_bot | I appreciate your answer, though I'm not sure how it addresses my question... | |
| Sep 17, 2016 at 21:46 | history | edited | HeinrichD | CC BY-SA 3.0 |
added 143 characters in body
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| Sep 17, 2016 at 21:40 | history | answered | HeinrichD | CC BY-SA 3.0 |