Talk:Constant sheaf

mathoverflow comment

The general definition of ${\underline {S}}$ (last paragraph) is wrong. Check out this mathoverflow question.

Also I think that the example is far too long ... but this is just my opinion. -oo- (talk) 10:30, 1 May 2010 (UTC)

nine inclusions?

In the "detailed example" why are there nine inclusions, and not five? 85.250.40.219 (talk) 23:04, 14 January 2011 (UTC)

Thanks for pointing that out. It appears that when this text was originally written (back in December 2007, see this diff), the editor was counting the trivial inclusions as well (such as {p} ⊆ {p}). In July 2008, a minor rewrite introduced the term "non-trivial" without changing the number from 9 down to 5 (see this diff). I'll fix it now. RobHar (talk) 23:34, 14 January 2011 (UTC)

separated presheaf?

The text claims that F is a separated presheaf and satisfies the gluing axiom, but not a sheaf. I believe the usual definition of a "separated presheaf" is simply one which satisfies the identity axiom (if two sections of U agree on all the members of some open cover of U then they are the same), not a sheaf whose restriction maps are injective (this almost never happens!). Thus any separated presheaf which satisfies the gluing axiom is a sheaf... 2620:101:F000:700:223:6CFF:FE98:4D1 (talk) 02:57, 6 August 2013 (UTC)

This is not correct. A separated presheaf is indeed one with injective restriction maps, and yes, they do occur in practice. For example, see the construction of the sheaf associated to a presheaf in SGA 4; Grothendieck and Verdier construct a functor L from presheaves to presheaves which turns all presheaves into separated presheaves and all separated presheaves into sheaves; thus applying L twice gives the sheaf associated to a presheaf. Ozob (talk) 03:28, 6 August 2013 (UTC)

Your response is not consistent with the definition provided at Sheaf (mathematics). 2620:101:F000:700:223:6CFF:FE98:4D1 (talk) 14:31, 8 September 2013 (UTC)

You're correct; I wrote that too quickly. I meant to say that having injective restriction maps implies that the presheaf is separated. Recall that ${\mathcal {F}}$ is a sheaf if for every open cover U_i of U, the sequence:

${\mathcal {F}}(U)\to \prod _{i}{\mathcal {F}}(U_{i})\rightrightarrows \prod _{i,j}{\mathcal {F}}(U_{i}\cap U_{j})$

is exact (stated in different terminology, the first arrow is an equalizer. Well, ${\mathcal {F}}$ is a separated presheaf if the first arrow is injective; if each restriction map is injective then the injectivity of the first arrow follows immediately.

I don't think the article is as clear as possible about this point, so I'll edit it a little. Ozob (talk) 19:38, 8 September 2013 (UTC)

To correct a subtle error: injectivity of ${\mathcal {F}}(U)\to \prod _{i\in I}{\mathcal {F}}(U_{i})$ is guaranteed from injectivity of each restriction map only if the index set is nonempty or if ${\mathcal {F}}(U)$ has at most one element anyway. This point already is clarified in the rest of the paragraph, so I will edit the beginning to match. — Preceding unsigned comment added by 132.229.30.176 (talk) 15:04, 5 February 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Constant sheaf/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

What is the normalization axiom? I don't find it anywhere. None of the books I've read about sheaves seem to mention it, and I can't find it online either. Carmen

Last edited at 15:43, 11 December 2008 (UTC). Substituted at 01:55, 5 May 2016 (UTC)

Question

By the stalks are "equal", does that mean they are literally equal? or up to an isomorphism? ${\mathcal {F}}_{x}={\mathcal {F}}_{y}$? Or ${\mathcal {F}}_{x}\simeq {\mathcal {F}}_{y}$? I think it should be the latter (it's similar to the change of base point functor) If so, the article needs to be clarified. -- Taku (talk) 18:35, 9 July 2017 (UTC)

Coproduct vs categorical product

@Jakob.scholbach thank you for your excellent detailed example on this page. It looks like it was migrated from Sheaf (mathematics) to this page based on looking at history.

I'm wondering if the use of ⊕ rather than ×/⊗ in the last part of the example is still correct. I read it as a coproduct, making me think of the Disjoint union (topology). Can the disjoint union hold enough information, though? Let's say we have the element (7,b) in ℤ⊕ℤ (where the b indicates that the number comes from the second ℤ set). Where would each of the projection morphisms map this? It seems like only one of the morphisms would be able to; the second wouldn't have enough information.

If you do think this should be changed then I can fix up the diagram tied to the text (I know that can be a headache), unless you want to. I'm obviously not certain that I'm understanding everything correctly, however. Davidvandebunte (talk) 18:20, 6 December 2023 (UTC)

@Davidvandebunte: The text and picture, as they stand, are correct, since in the category of abelian groups, the coproduct and product agree (unlike in, say, the category of sets, where the coproduct is the disjoint union). However, there is a (minor) point to be made to replace the direct sum symbol by the product, since the sheaf axiom generally requires a disjoint union of spaces to be mapped to a product (of sets, or abelian groups, etc.). Feel free to change it if you want. Jakob.scholbach (talk) 13:10, 27 December 2023 (UTC)

I've added an explanation that '\otimes' means product here as I've never seen that notation before and spent fruitless hours trying to see if you could do tensor products of integers. Was there an important reason not to just use '\times' like all the other wikipedia pages that use category theoretical products? Because I'd love to just change it to '\times' but won't do that on my own.Brirush (talk) 23:16, 27 May 2024 (UTC)

@Brirush Feel free to change it to '\times' to fit the other articles. Davidvandebunte (talk) 20:06, 1 June 2024 (UTC)