Timeline for Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2023 at 15:07 | history | edited | Joscha Gillessen | CC BY-SA 4.0 |
added 2 characters in body
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| Apr 22, 2023 at 2:28 | comment | added | Zhen Lin | I believe KS is considered unorthodox. I don't often see it recommended as an introduction or as a reference. | |
| Apr 22, 2023 at 2:25 | comment | added | Joscha Gillessen | I don’t have a feeling on what is considered usual notation. I assumed, that the conventions of KS were standard. What would you recommend me to clarify? | |
| Apr 22, 2023 at 2:23 | comment | added | Zhen Lin | That is a very unusual way of doing things... but yes, then it is true. You should clarify your notation. | |
| Apr 22, 2023 at 2:22 | comment | added | Joscha Gillessen | In the original question it was written “ To simplify, let me consider only set-valued presheaves.” As my argument does not need this simplification, I figured, it would be appropriate to say “ where the presheaves are allowed to take values in any category $\mathcal{A}$ with enough limits” | |
| Apr 22, 2023 at 2:18 | comment | added | Joscha Gillessen | Here $F$ is considered to be extended to a contra variant functor $Psh(C,\operatorname{Set}) \rightarrow \mathcal{A}$ as described on Page 408 of KS, line 6. $F$ then commutes with small limits. | |
| Apr 22, 2023 at 2:16 | comment | added | Zhen Lin | You wrote: "where the presheaves are allowed to take values in any category $\mathcal{A}$ with enough limits". If you don't work in that generality then there is no need to write it. | |
| Apr 22, 2023 at 2:12 | comment | added | Joscha Gillessen | Where do I need this assumption on $\mathcal{A}$? The factorisation of $f$ is done in the category $Psh(C,\operatorname{Set})$, not $\mathcal{A}$. | |
| Apr 21, 2023 at 22:30 | comment | added | Zhen Lin | You are being careless about your hypotheses. Clearly you are assuming that $\mathcal{A}$ has some kind of epi–mono factorisation. You seem also make the common mistake of equating the sheaf condition to preservation of limits. | |
| S Apr 21, 2023 at 14:36 | review | First answers | |||
| Apr 21, 2023 at 14:37 | |||||
| S Apr 21, 2023 at 14:36 | history | edited | LSpice | CC BY-SA 4.0 |
Links and tidying
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| Apr 21, 2023 at 13:25 | review | Late answers | |||
| Apr 21, 2023 at 13:35 | |||||
| S Apr 21, 2023 at 13:09 | review | First answers | |||
| Apr 21, 2023 at 13:21 | |||||
| S Apr 21, 2023 at 13:09 | history | answered | Joscha Gillessen | CC BY-SA 4.0 |