Timeline for What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?
Current License: CC BY-SA 2.5
7 events
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| Oct 16, 2009 at 7:10 | comment | added | user332 | It's not clear to me at all how to make this translation. It seems that if you replace stalks with equivalence classes of representatives pointwise, then you are going to run into trouble trying to get them to match as a cover over U. | |
| Oct 16, 2009 at 2:17 | comment | added | S. Carnahan♦ | I believe if you translate Hartshorne's construction of his + functor, by replacing elements of stalks with equivalence classes of representing pairs (U,s), you get everyone else's + functor. Taking H^0 and passing to the limit of refinements is the same as taking compatible functions to the union of stalks. The compatibility for these functions becomes interesting exactly when you have a cover by disjoint open sets. (Note also the bizarre statement of Hartshorne Chapter 2, exercise 1.1.) | |
| Oct 15, 2009 at 19:57 | comment | added | Anton Geraschenko | I don't know an example that doesn't use ∅ in this sneaky way. I think it's silly to assume that a presheaf sends ∅ to the final object; I prefer to just say a presheaf is a functor from the topology (thought of as a category) to some other category. For a sheaf, the sheaf axiom already implies that ∅ goes to the final object. | |
| Oct 15, 2009 at 19:55 | comment | added | user332 | I think the key point behind the + functor is that it allows us to do sheafification without referring to stalks, which is something we may not have for a general Grothendieck topology. As I recall, Hartshorne's construction of the sheafification is different and makes use of stalks in an essential way. In any case, I don't think that concerns about P(∅) tell the full story. I'm not satisfied with the above example because I suspect that there is another reason why P^+ can fail to be a sheaf which has nothing to do with P(∅) not being a point. | |
| Oct 15, 2009 at 19:44 | comment | added | Reid Barton | Is this example why some texts (such as Hartshorne) require a presheaf to send ∅ to the final object? That always bugged me. But I guess it simplifies the construction of the sheafification (when your Grothendieck topology comes from a topological space). | |
| Oct 15, 2009 at 19:39 | comment | added | user332 | This seems to work, but it wasn't really what I had in mind. I was hoping for something like a presheaf which stays nonseparated as we restrict to arbitrarily small open sets. Suppose we revise the question and assume that P(∅) = one point. Can we find an example then? | |
| Oct 15, 2009 at 19:12 | history | answered | Anton Geraschenko | CC BY-SA 2.5 |