The Stacks project

Remark 6.7.2. There is always a bit of confusion as to whether it is necessary to say something about the set of sections of a sheaf over the empty set $\emptyset \subset X$. It is necessary, and we already did if you read the definition right. Namely, note that the empty set is covered by the empty open covering, and hence the “collection of sections $s_ i$” from the definition above actually form an element of the empty product which is the final object of the category the sheaf has values in. In other words, if you read the definition right you automatically deduce that $\mathcal{F}(\emptyset ) = \textit{a final object}$, which in the case of a sheaf of sets is a singleton. If you do not like this argument, then you can just require that $\mathcal{F}(\emptyset ) = \{ *\} $.

In particular, this condition will then ensure that if $U, V \subset X$ are open and disjoint then

\[ \mathcal{F}(U \cup V) = \mathcal{F}(U) \times \mathcal{F}(V). \]

(Because the fibre product over a final object is a product.)


Comments (4)

Comment #3427 by on

Small typo: There's an "s" missing at the end of the word "section" in "... and hence the 'collection of section ' from the definition above..."

Comment #5395 by Peter Johnson on

I think the explanation (using the empty product) is not quite right. What the argument really depends on is that in Set, or the usual targets, EVERY 1-element object is terminal.

In a more general target category (of sets with structure, certain morphisms), it can fail. Every sheaf gives rise to a counter-example after adjusting the target category by making a new isomorphic copy 0 of the final object, then removing all morphisms going out of 0.

However, the idea used here, of passing through an empty product (which supposes the category has this) does in fact a strengthen the definition enough to rule out bad examples. The above 'descent' property (1) may well best be replaced by the usual "element-free" one involving equalizers of two morphisms between products, if you don't need to go beyond complete categories.

Comment #5626 by on

For those who are interested, let me point out that sheaves of sets with algebraic structure are discussed in Section 6.9. However, instead of reading that section, you should just go ahead and look for (a possible) definition of sheaves with values in any category given in the chapter on sites in Section 7.7 (specifically Definition 7.7.6).


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