Timeline for A very elementary question on the definition of sheaf on a site
Current License: CC BY-SA 4.0
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| Nov 30, 2020 at 14:46 | comment | added | Simon Henry | You can simply take the category of Sets (i.e., $U_i$, $V$ and $Z$ are all just sets), that should make everything clear. The general picture as I describe it informally always make sense when you work in the topos of sheaves instead of the site, but as a topos is basically "a category that behave like the category of set" looking at what happen in the category of sets should be illuminating enough. | |
| Nov 30, 2020 at 14:34 | comment | added | gualterio | @Simon Henry So the condition for i=j has its own meaning. As I'm not completely easy with category theory, I don't perfectly understand your explanation on the equivalence relation. Can you suggest some 'general' class of categories where your explanation is more explicit and complete? | |
| Nov 30, 2020 at 14:29 | vote | accept | gualterio | ||
| Nov 30, 2020 at 14:18 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 30, 2020 at 14:12 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 30, 2020 at 14:06 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 30, 2020 at 13:48 | comment | added | Achim Krause | Wait, the two restrictions to $U_i\times_U U_i$ can still be different, as the two projections $U_i\times_U U_i\to U_i$ don't necessarily agree (only if the maps $U_i\to U$ are monomorphisms). This doesn't come up in the small site of a topological space, but definitely in the context of more general sites and etale cohomology. | |
| Nov 30, 2020 at 13:42 | history | answered | Simon Henry | CC BY-SA 4.0 |